Kelvin–Voigt equations with anisotropic diffusion,relaxation and damping: Blow-up and large time behavior

Abstract

A nonlinear initial and boundary-value problem for the Kelvin–Voigt equations with anisotropic diffusion, relaxation and absorption/damping terms is considered in this work. The global and local unique solvability of the problem was established in (J. Math. Anal. Appl. 473(2) (2019) 1122–1154). In the present work, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients should interact with the problem data for the solutions of this problem display exponential and polynomial time-decays. We also establish the conditions for the solutions of this problem to blow-up in a finite time in three different cases: problem without convection, full anisotropic problem, and the problem with isotropic relaxation.

Keywords

Kelvin–Voigt equations anisotropic diffusion anisotropic relaxation anisotropic damping large time behavior blow-up

1. Introduction

Kelvin–Voigt equations are used in the applications to model the response of materials that exhibit all intermediate range of properties between an elastic solid and a viscous fluid, as, for instance, polymers. Simple constitutive relations describing the behaviour of these materials go back to the works of Maxwell [23] and Kelvin [30], and are usually obtained by combining Hook’s law of linear elasticity with Newton’s law of viscosity (see e.g. [11]). Motivated by Kelvin’s work, Voigt [32] has derived a system of equations describing the behavior of elastic solids with viscous properties, which is known today by the names of Voigt equations, Kelvin equations, or Kelvin–Voigt equations. Later on, inspired in the works by Kelvin and Voigt, Oskolkov [25] has derived the following system of equations for homogeneous and incompressible fluids with elastic properties, $\begin{array}{l} (1.1) & div u = 0, \\ (1.2) & ρ (\frac{\partial u}{\partial t} + (u \cdot \nabla) u) = ρ f - \nabla π + μ △ u + κ △ u_{t}, \end{array}$ where u denotes the velocity field, π is the pressure, ρ is the constant density, f is the external forces field, κ denotes the relaxation time and μ is the fluid viscosity. Oskolkov denoted the system (1.1)–(1.2) as the Kelvin–Voigt equations for fluid flows with elastic properties, but it should be noted that neither Kelvin nor Voigt have suggested any stress-strain relation or system of governing equations for viscoelastic fluids. Mathematically speaking, the main feature of the Kelvin–Voigt equations (1.2), is that the term $κ △ u_{t}$ works as a regularization of the Navier–Stokes equations in the sense that the correspondingly incompressible initial and boundary-value 3d problem has a unique global solution [18,26]. These equations have also been proposed as a good model for numerical simulations of the Navier–Stokes equations, provided $κ > 0$ is small [12]. Kelvin–Voigt equations of the type (1.1)–(1.2) have been already considered in different settings, including with viscous or relaxation terms described by different power-laws, or considered with an extra damping/absorption term [8,9,33]. The present paper is a continuation of the work [4], where we have considered the initial and boundary-value problem posed by the following full anisotropic Kelvin–Voigt equations perturbed by anisotropic relaxation, diffusion and damping, $\begin{array}{l} (1.3) & \begin{matrix} \partial_{t} u + (u \cdot \nabla) u = & f - \nabla π + \sum_{i = 1}^{d} D_{i} (κ_{i} | D_{i} u |^{q_{i} - 2} \partial_{t} (D_{i} u) + ν_{i} | D_{i} u |^{p_{i} - 2} D_{i} u) \\ + \sum_{i = 1}^{d} γ_{i} | u |^{m_{i} - 2} u, in Ω \times (0, T], \end{matrix} \\ (1.4) & div u = 0 in Q_{T}, \\ (1.5) & u = u_{0} in Ω for t = 0, \\ (1.6) & u = 0 on Γ_{T}, \end{array}$ in a cylinder $Q_{T} : = Ω \times [0, T]$ , where its boundary is denoted by $Γ_{T} : = \partial Ω \times [0, T]$ . Here, $Ω \subset R^{d}$ , $d ⩾ 2$ , is a bounded domain with its boundary denoted by $\partial Ω$ , and $T > 0$ is some fixed time. The vector function $u = (u_{1}, \dots, u_{d})$ and the scalar function π are the unknowns of problem, whereas the vector function $f = (f_{1}, \dots, f_{d})$ is a given datum. The notation $D_{i} u$ is understood as the partial derivative of u with respect to $x_{i}$ : $D_{i} u = (\partial_{i} u_{1}, \dots, \partial_{i} u_{d})$ , where $\partial_{i} u_{j} = \frac{\partial u_{j}}{\partial x_{i}}$ for any $i, j \in {1, \dots, d}$ . The exponents $q_{i}$ , $p_{i}$ and $m_{i}$ are assumed to be constant with possible distinct values and such that $1 < q_{i}, p_{i}, m_{i} < \infty$ for all $i \in {1, \dots, d}$ , whereas $κ_{i}$ , $ν_{i}$ and $γ_{i}$ are possible distinct constants such that $\begin{matrix} κ_{i} ⩾ 0, - \infty < γ_{i} < \infty, ν_{i} > 0 \forall i \in {1, \dots, d} . \end{matrix}$ In the scope of Fluid Mechanics, the extra term $\sum_{i = 1}^{d} γ_{i} | u |^{m_{i} - 2} u$ in the equation (1.4) accounts for a sink or a source within the system. If $γ_{i} < 0$ , this term describes a sink in the direction of $x_{i}$ , whereas if $γ_{i} > 0$ it depicts a source in the direction of $x_{i}$ . Interestingly is the fact that we may have sinks in some directions and sources in another directions.

In [4], we have established the conditions for the global and local existence of weak solutions to the problem (1.3)–(1.6), and for the uniqueness as well. Here, we proceed our analysis for the problem (1.3)–(1.6), specifically by studying the large time behavior of the solutions in the case $γ_{i} < 0$ for all $i \in {1, \dots, d}$ (with anisotropic absorption term). In the case of $γ_{i} > 0$ for all $i \in {1, \dots, d}$ (with anisotropic damping term), we study the blow-up in a finite time of the solutions to the problem (1.3)–(1.6).

When $q_{i} = 2$ , $p_{i} = 2$ , $κ_{i} = κ$ , $ν_{i} = ν$ and $γ_{i} = 0$ for all $i \in {1, \dots, d}$ , equations (1.3)–(1.4) reduce to the Kelvin–Voigt model (1.1)–(1.2) studied by Oskolkov [26] and Ladyzhenskaya [18]. The case of $κ_{i} = 0$ for all $i \in {1, \dots, d}$ , corresponds to the anisotropic Navier–Stokes equations studied by Antontsev and Oliveira [1–3,13]. If in addition to $ν_{i} = ν$ , $p_{i} = p$ , $γ_{i} = γ < 0$ and $m_{i} = m$ , we have $κ_{i} = κ$ and $q_{i} = q$ , we fall in the generalized Kelvin–Voigt with damping worked out by Antontsev and Khompysh [8,9]. On the other hand, if, in addition to $κ_{i} = 0$ , $ν_{i} = ν$ , $p_{i} = p$ , we also have $γ_{i} = 0$ for all $i \in {1, \dots, d}$ , we get the generalized Navier–Stokes equations that govern incompressible power-law fluids and which have been studied during the last 50 years, after the pioneer works by Ladyzhenskaya [19] and Lions [21].

In recent years, a sort of damped Navier–Stokes equations are being used, in the applications, to model certain porous media flows, where the damping terms account for the Darcy and Forchheimer drags due to the porous matrix (see e.g. [24]). Damped Navier–Stokes equations appear also as a regularization technique to prove existence results for generalized Navier–Stokes equations that govern flows of generalized Newtonian fluids (see e.g. [15]). But, the main feature of damped Navier–Stokes equations is that, due to the damping term, one can proves some qualitative properties, as the confinement of the solutions in some subdomain or its extinction in a finite time, that otherwise would be impossible to obtain (see e.g. [1,5,6]).

This work is organized as follows. In Section 2, we introduce the anisotropic function spaces with which we will work with and we recall some auxiliary lemmas that will be used in the sequel and give the definition of weak solutions. The energy inequalities are proved in Section 3. In Sections 4 we establish polynomial and exponential time-decay properties for the weak solutions to the problem (1.3)–(1.6) with anisotropic absorption terms (with $γ_{i} < 0$ , $i = 1, 2, \dots, d$ ). The blow-up of the solutions to the problem (1.3)–(1.6) with anisotropic damping terms (with $γ_{i} > 0$ , $i = 1, \dots, d$ ) is proved in Section 5, in the case without the convective term. In the general case, with the convective term, the blow-up is established in Section 6. The case of linear isotropic relaxation, for which existence of weak solutions is known, is studied in Section 7.

The notation used throughout this article and the main notions of the considered (isotropic) function spaces are largely standard in the literature of Partial Differential Equations and in Mathematical Fluid Mechanics as well. We address the reader e.g. to the monographs [21,22,29] for any question related to that matter.

2. Preliminaries

2.1. Anisotropic function spaces

The presence of different exponents $q_{i}$ , $p_{i}$ and $m_{i}$ for distinct directions, lead us to look for the solutions to the problem (1.3)–(1.6) in some anisotropic Sobolev space. We define the vectors q, p and m in $R^{d}$ , whose components are the exponents of the anisotropic terms considered in (1.3), by $\begin{matrix} q : = (q_{1}, \dots, q_{d}), p : = (p_{1}, \dots, p_{d}), m : = (m_{1}, \dots, m_{d}), \end{matrix}$ with $1 < q_{i}, p_{i}, m_{i} < \infty$ for all $i \in {1, \dots, d}$ . To avoid any misunderstand that q, p and m are in fact multi-component, we shall emphasize this by writing $\vec{q}$ , $\vec{p}$ and $\vec{m}$ , respectively, in the rest of our work. For the sake of simplifying the notation, we assume throughout the text that the components of $\vec{q}$ , $\vec{p}$ and $\vec{m}$ satisfy $\begin{array}{l} (2.1) & q_{1} ⩽ q_{2} ⩽ \dots ⩽ q_{d}, \\ (2.2) & p_{1} ⩽ p_{2} ⩽ \dots ⩽ p_{d}, \\ (2.3) & m_{1} ⩽ m_{2} ⩽ \dots ⩽ m_{d} . \end{array}$

Considering the unidirectional Sobolev spaces $\begin{matrix} W_{i}^{1, p_{i}} (Ω) : = {v \in W^{1, 1} (Ω) \cap L^{p_{1}} (Ω) : D_{i} v \in L^{p_{i}} (Ω)}, i \in {1, \dots, d}, \end{matrix}$ which are Banach spaces for the norms $\begin{matrix} (2.4) & ‖ v ‖_{W_{i}^{1, p_{i}} (Ω)} : = ‖ v ‖_{L^{p_{1}} (Ω)} + ‖ D_{i} v ‖_{L^{p_{i}} (Ω)}, i \in {1, \dots, d}, \end{matrix}$ we define the following anisotropic Sobolev space $\begin{matrix} W^{1, \vec{p}} (Ω) : = ⋂_{i = 1}^{d} W_{i}^{1, p_{i}} (Ω), \end{matrix}$ with the norm defined by $\begin{matrix} (2.5) & ‖ v ‖_{W^{1, \vec{p}} (Ω)} : = ‖ v ‖_{L^{p_{1}} (Ω)} + \sum_{i = 1}^{d} ‖ D_{i} v ‖_{L^{p_{i}} (Ω)} . \end{matrix}$

An important limitation of the anisotropic Sobolev space $W^{1, \vec{p}} (Ω)$ , is that, for bounded domains Ω, the validity of the Sobolev imbeddings is restricted to rectangular domains (see e.g. [16]). In fact, for rectangular domains Ω, the following imbedding is continuous (cf. [28, Theorem 1]) $\begin{matrix} (2.6) & W^{1, \vec{p}} (Ω) ↪ L^{s} (Ω) for any s : \{\begin{matrix} 1 ⩽ s ⩽ {\overline{p}}^{*}, & \sum_{j = 1}^{d} \frac{1}{p_{j}} > 1 \\ 1 ⩽ s < \infty, & \sum_{j = 1}^{d} \frac{1}{p_{j}} ⩽ 1, \end{matrix} \end{matrix}$ where ${\overline{p}}^{*}$ denotes the Sobolev conjugate of $\overline{p}$ , the harmonic mean of $p_{1}, \dots, p_{d}$ : $\begin{matrix} {\overline{p}}^{*} : = \frac{d \overline{p}}{d - \overline{p}}, \overline{p} : = \frac{d}{\sum_{j = 1}^{d} \frac{1}{p_{j}}}, \sum_{j = 1}^{d} \frac{1}{p_{j}} > 1 \Rightarrow {\overline{p}}^{*} = \frac{d}{\sum_{j = 1}^{d} \frac{1}{p_{j}} - 1} . \end{matrix}$ Moreover, the imbedding (2.6) is compact (cf. [28, Theorem 2]), and we denote this fact by writing $\begin{matrix} (2.7) & W^{1, \vec{p}} (Ω) ↪ ↪ L^{s} (Ω) for any s : 1 ⩽ s < {\overline{p}}^{*} . \end{matrix}$ In some situations, not only it is possible to remove the restrictions on the shape of the domain, but it is also feasible to enlarge the interval of s for the validity of (2.6) and (2.7). Let us see this fact by defining $\begin{matrix} W_{0}^{1, \vec{p}} (Ω) : = closure of C_{0}^{\infty} (Ω) in the anisotropic norm of W^{1, \vec{p}} (Ω) . \end{matrix}$ In this case, we have (cf. [14, Theorem 1]), in the interesting case of $\sum_{j = 1}^{d} \frac{1}{p_{j}} > 1$ , that $\begin{matrix} (2.8) & \begin{array}{l} W_{0}^{1, \vec{p}} (Ω) ↪ L^{s} (Ω) for any s : 1 ⩽ s ⩽ p_{a}^{*}, \\ W_{0}^{1, \vec{p}} (Ω) ↪ ↪ L^{s} (Ω) for any s : 1 ⩽ s < p_{a}^{*}, \end{array} \end{matrix}$ where $p_{a}^{*}$ is the critical exponent defined by $\begin{matrix} p_{a}^{*} : = max {{\overline{p}}^{*}, p_{d}} . \end{matrix}$

Remark 2.1.
Note that, for $d = 2$ , ${\overline{p}}^{} > p_{d}$ and therefore $p_{a}^{} = {\overline{p}}^{}$ . But, if $d > 2$ , it may well happen that $p_{d} > {\overline{p}}^{}$ . In fact, for $d > 2$ , $\begin{matrix} p_{d} > {\overline{p}}^{} \Leftrightarrow \overline{p} < \frac{p_{d} d}{p_{d} + d} \Leftrightarrow \sum_{j = 1, p_{j} \neq p_{d}}^{d} \frac{1}{q_{j}} > 1 + \frac{d - 1}{p_{d}} . \end{matrix}$ This means that, in typical situations when $d > 2$ , $p_{a}^{} = p_{d}$ if, at least, one of the components $p_{i}$ of $\vec{p}$ is too far apart from the others. For instance, when $d = 3$ , $\vec{p} = (\frac{10}{7}, \frac{20}{7}, p_{d})$ , we have $p_{d} > {\overline{p}}^{}$ if and only if $p_{d} > 40$ . In this example, the first two components $p_{1}$ and $p_{2}$ are relatively close to each other, but we may have all the three components $p_{1}$ , $p_{2}$ and $p_{3}$ sufficiently far apart one from each other, as shows the example $\vec{p} = (\frac{1001}{1000}, \frac{1001}{2}, p_{d})$ for which $p_{d} > {\overline{p}}^{}$ if and only $p_{d} > 2002$ .

2.2. Auxiliary results

As a particular case of (2.8), it can be derived the following result, originally proved by Troisi [31] and later on extended in [14,27].

Lemma 2.1.
Let $Ω \subset R^{d}$ be an open bounded domain with a Lipschitz-continuous boundary $\partial Ω$ . Then for any $u \in W_{0}^{1, \vec{p}} (Ω)$ $\begin{matrix} (2.9) & ‖ u ‖_{L^{s} (Ω)} ⩽ C {(\prod_{i = 1}^{d} ‖ D_{i} u ‖_{L^{p_{i}} (Ω)})}^{\frac{1}{d}} \end{matrix}$ for $s \in [1, p_{a}^{}]$ if $\sum_{i = 1}^{d} \frac{1}{p_{i}} > 1$ , or* $s \in [1, \infty)$ otherwise, and where $C = C (Ω, d, s, p_{i})$ is a positive constant.
Proof.
The proof combines [31, Theorem 1.2], [27, Theorem 3.1] and [14, Theorem 1]. □

Next, we recall two lemmas which are very important in the proofs of the results we shall establish in the sequel.
Lemma 2.2.
Suppose that a positive and differentiable function $Φ (t)$ satisfies the following nonlinear differential inequality, $\begin{matrix} Φ^{'} (t) + C Φ^{α} (t) ⩽ 0 \forall t > 0, \end{matrix}$ for some positive constants α and C. Then the following assertions are valid:
If $α \in (0, 1)$ , then $\begin{matrix} 0 ⩽ Φ (t) ⩽ {({Φ (0)}^{1 - α} - (1 - α) C t)}_{+}^{\frac{1}{1 - α}} \forall t ⩾ 0, \end{matrix}$ where $f_{+} : = max {0, f}$ . This means, $\begin{matrix} \exists T_{max} = \frac{{Φ (0)}^{1 - α}}{(1 - α) C} > 0 : Φ (t) = 0 \forall t ⩾ T_{max}; \end{matrix}$

If $α = 1$ , then $\begin{matrix} Φ (t) ⩽ Φ (0) e^{- C t} \forall t ⩾ 0; \end{matrix}$

If $α > 1$ , then $\begin{matrix} Φ (t) ⩽ Φ (0) {(1 + t C (α - 1) Φ {(0)}^{α - 1})}^{- \frac{1}{α - 1}} \forall t ⩾ 0 . \end{matrix}$

Proof.
See e.g. [7, Chapter 2]. □

The proof of blow-up of the solutions to the problem (1.3)–(1.6) is based on the following lemma.
Lemma 2.3.
Suppose that a positive and twice-differentiable function $Φ (t)$ satisfies the following nonlinear differential inequality, $\begin{matrix} (2.10) & Φ^{″} (t) Φ (t) - (1 + α) {(Φ^{'} (t))}^{2} ⩾ - 2 B_{1} Φ^{'} (t) Φ (t) - B_{2} Φ^{2} (t) \forall t > 0, \end{matrix}$ for some constants $α > 0$ and $B_{1}, B_{2} ⩾ 0$ .

If $\begin{matrix} (2.11) & Φ (0) > 0, Φ^{'} (0) > - γ_{2} α^{- 1} Φ (0), and B_{1} + B_{2} > 0, \end{matrix}$ then $\begin{matrix} (2.12) & \exists t_{1} > 0 : Φ (t) \to + \infty, as t \to t_{1} ⩽ \frac{1}{2 \sqrt{B_{1}^{2} + α B_{2}}} ln (\frac{γ_{1} Φ (0) + α Φ^{'} (0)}{γ_{2} Φ (0) + α Φ^{'} (0)}), \end{matrix}$ where $γ_{1} = - B_{1} + \sqrt{B_{1}^{2} + α B_{2}}$ , $γ_{2} = - B_{1} - \sqrt{B_{1}^{2} + α B_{2}}$ .
Proof.
See e.g. [17, Lemma 1.1] and [20, Theorem I]. □

For the proofs of some results established in this work, we will make use of the following version of Young’s inequality, $\begin{matrix} (2.13) & a b ⩽ \frac{ϵ^{p} a^{p}}{p} + \frac{b^{q}}{q ϵ^{q}}, a, b > 0, ϵ > 0, 1 < p, q < \infty, \frac{1}{p} + \frac{1}{q} = 1 . \end{matrix}$ In the case of $p = q = 2$ and $ϵ = \sqrt{ε}$ in (2.13), we will refer to this case as Cauchy’s inequality with ε.
2.3. Weak formulation

We recall the classical function spaces of Mathematical Fluid Mechanics, $\begin{array}{l} V : = {v \in C_{0}^{\infty} (Ω) : div v = 0}, \\ (2.14) & H : = closure of V in the norm of L^{2} (Ω), \\ V_{p} : = closure of V in the norm of W^{1, p} (Ω) . \end{array}$ We define the anisotropic analogue of $V_{p}$ as follows, $\begin{matrix} (2.15) & V_{\vec{p}} : = ⋂_{i = 1}^{d} V_{p_{i}}, \end{matrix}$ where, similarly to (2.14), $\begin{matrix} V_{p_{i}} : = closure of V in the norm of W_{i}^{1, p_{i}} (Ω) . \end{matrix}$ We introduce the parabolic anisotropic spaces, $\begin{matrix} L^{\vec{p}} (Q_{T}) : = ⋂_{i = 1}^{d} L^{p_{i}} (Q_{T}), L^{\vec{m}} (Q_{T}) : = ⋂_{i = 1}^{d} L^{m_{i}} (Q_{T}), \end{matrix}$ which are Banach spaces for the norms defined by $\begin{matrix} ‖ v ‖_{L^{\vec{p}} (Q_{T})} : = \sum_{i = 1}^{d} ‖ v ‖_{L^{p_{i}} (Q_{T})}, ‖ v ‖_{L^{\vec{m}} (Q_{T})} : = \sum_{i = 1}^{d} ‖ v ‖_{L^{m_{i}} (Q_{T})} . \end{matrix}$ Now, we consider the unidirectional Bochner space, $\begin{matrix} L^{p_{i}} (0, T; V_{p_{i}}) : = {v : [0, T] \to V_{p_{i}} ∣ v \in L^{p_{1}} (Q_{T}) \land D_{i} v \in L^{p_{i}} (Q_{T}) \forall i \in {1, \dots, d}}, \end{matrix}$ which is a Banach space for the norm $\begin{matrix} ‖ v ‖_{L^{p_{i}} (0, T; V_{p_{i}})} : = ‖ v ‖_{L^{p_{1}} (Q_{T})} + ‖ D_{i} v ‖_{L^{p_{i}} (Q_{T})} . \end{matrix}$ Then we define the anisotropic Banach space $\begin{matrix} (2.16) & L^{\vec{p}} (0, T; V_{\vec{p}}) : = ⋂_{i = 1}^{d} L^{p_{i}} (0, T; V_{p_{i}}), \end{matrix}$ whose norm is defined by $\begin{array}{rcl} ‖ v ‖_{L^{\vec{p}} (0, T; V_{\vec{p}})} : = ‖ v ‖_{L^{p_{1}} (Q_{T})} + \sum_{i = 1}^{d} ‖ D_{i} v ‖_{L^{p_{i}} (Q_{T})} . \end{array}$

We are now in conditions to define the notion of weak solutions to the problem (1.3)–(1.6) we are interested in.

Definition 2.1.
Let $d ⩾ 2$ and $\vec{q} = (q_{1}, \dots, q_{d})$ , $\vec{p} = (p_{1}, \dots, p_{d})$ and $\vec{m} = (m_{1}, \dots, m_{d})$ , with $1 < q_{i}, p_{i}, m_{i} < \infty$ for all $i \in {1, \dots, d}$ . Assume that $f \in L^{ρ^{'}} (Q_{T})$ , where $ρ^{'} = \frac{ρ}{ρ - 1}$ and $ρ : = min {2, p_{1}, m_{1}}$ . A vector field u is a weak solution to the problem (1.3)–(1.6), if:
$u \in L^{\infty} (0, T; H \cap V_{\vec{q}}) \cap L^{\vec{p}} (0, T; V_{\vec{p}}) \cap L^{\vec{m}} (Q_{T})$ ;

$u (0) = u_{0}$ ;

For every $v \in H \cap V_{\vec{q}} \cap V_{\vec{p}} \cap L^{\vec{m}} (Ω) \cap L^{θ} (Ω)$ $\begin{matrix} (2.17) & \begin{matrix} \frac{d}{d t} (\int_{Ω} u (t) \cdot v d x + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i} - 1} \int_{Ω} {| D_{i} u (t) |}^{q_{i} - 2} D_{i} u (t) \cdot D_{i} v d x) \\ + \sum_{i = 1}^{d} ν_{i} \int_{Ω} {| D_{i} u (t) |}^{p_{i} - 2} D_{i} u (t) \cdot D_{i} v d x + \int_{Ω} [(u (t) \cdot \nabla) u (t)] \cdot v d x \\ = \int_{Ω} f (t) \cdot v d x + \sum_{i = 1}^{d} γ_{i} \int_{Ω} {| u (t) |}^{m_{i} - 2} u (t) \cdot v d x, \end{matrix} \end{matrix}$ in the distribution sense on $(0, T)$ , and where θ is defined by $\begin{matrix} \frac{1}{p_{a}^{}} + \frac{1}{p_{1}} + \frac{1}{θ} = 1 . \end{matrix}$

As usual, condition $u (0) = u_{0}$ is interpreted in the following sense: $\begin{matrix} lim_{t \to 0^{+}} \int_{Ω} u (t) \cdot v d x = \int_{Ω} u_{0} \cdot v d x \forall v \in H \cap V_{\vec{q}} \cap V_{\vec{p}} \cap L^{\vec{m}} (Ω) \cap L^{θ} (Ω) . \end{matrix}$

The assumption $v \in L^{θ} (Ω)$ is necessary to control the boundedness of the convective integral term when u merely belongs to $L^{\infty} (0, T; H \cap V_{\vec{q}}) \cap L^{\vec{p}} (0, T; V_{\vec{p}}) \cap L^{\vec{m}} (Q_{T})$ . But, if $θ ⩽ max {2, m_{1}, q_{a}^{}, p_{a}^{}}$ we just need to require that $v \in H \cap V_{\vec{q}} \cap V_{\vec{p}} \cap L^{\vec{m}} (Ω)$ , since, in this case, there hold the imbeddings $H ↪ L^{θ} (Ω)$ , $L^{\vec{m}} (Ω) ↪ L^{θ} (Ω)$ , $V_{\vec{q}} ↪ L^{θ} (Ω)$ and $V_{\vec{p}} ↪ L^{θ} (Ω)$ . In particular, if $θ ⩽ max {q_{a}^{}, p_{a}^{}}$ and $max {2, m_{d}} ⩽ max {q_{a}^{}, p_{a}^{*}}$ , we only need to require that $v \in V_{\vec{q}} \cap V_{\vec{p}}$ .

Note that, according to Riesz representation theorem, each element ϕ belonging to the dual space ${(L^{ρ} (Ω))}^{'}$ of $L^{ρ} (Ω)$ , where $1 < ρ < \infty$ , can be identified with exactly one $g \in L^{ρ^{'}} (Ω)$ , where $ρ^{'} = \frac{ρ}{ρ - 1}$ , such that $\begin{matrix} {⟨ ϕ, v ⟩}_{{(L^{ρ} (Ω))}^{'} \times L^{ρ} (Ω)} = \int_{Ω} g \cdot v d x \forall v \in L^{ρ} (Ω) \end{matrix}$ and $‖ ϕ ‖ = ‖ g ‖_{L^{ρ^{'}} (Ω)}$ . Throughout this work, the integral term of (2.17) involving the forces field shall be understood in this sense.

It should also be noted that assumption $f \in L^{ρ^{'}} (Q_{T})$ for $ρ = min {2, p_{1}, m_{1}}$ assures us that one of the alternatives holds: (i) $f \in L^{2} (Q_{T})$ ; (ii) $f \in L^{{\vec{p}}^{'}} (Q_{T})$ ; (iii) $f \in L^{{\vec{m}}^{'}} (Q_{T})$ ; where ${\vec{p}}^{'} = (p_{1}^{'}, \dots, p_{d}^{'})$ and ${\vec{m}}^{'} = (m_{1}^{'}, \dots, m_{d}^{'})$ are the d-uples with components $p_{i}^{'} = \frac{p_{i}}{p_{i} - 1}$ and $m_{i}^{'} = \frac{m_{i}}{m_{i} - 1}$ , respectively.
2.4. Previous results

The following global existence result has been established in [4, Theorem 7.1].

Theorem 2.1 (Global existence).

Let Ω be a bounded domain in $R^{d}$ , $d ⩾ 2$ , with a Lipschitz-continuous boundary $\partial Ω$ . Assume that $k_{i} = k$ and $q_{i} = q = 2$ for all $i \in {1, \dots, d}$ , $κ > 0$ , and $u_{0} \in V_{2}$ , $f \in L^{{\vec{p}}^{'}} (Q_{T})$ or $f \in L^{2} (Q_{T})$ . Assume one of the alternatives:

If $γ_{d} < 0$ , let us suppose that one and only one of the following conditions is verified, $\begin{array}{l} γ_{i} ⩽ 0 for all i \in {1, \dots, d - 1}, \\ m_{d} > m_{d - 1}, \\ \sum_{\begin{array}{c} i = 1 \\ γ_{i} ⩾ 0 \end{array}}^{d - 1} γ_{i} < - γ_{d} and m_{d} ⩾ m_{d - 1}; \end{array}$

If $γ_{d} > 0$ , let us suppose that one and only one of the following conditions is verified, $\begin{array}{l} m_{d} ⩽ 2 \\ 2 < m_{d} < \overline{p} . \end{array}$

In addition to the alternatives (1) and (2), let us assume that one and only one of the conditions in each one of the following two sets of hypotheses is verified,

\begin{array}{l} d ⩽ 4 \\ d > 4 and \overline{p} > 4, \end{array}

and

\begin{array}{l} d ⩽ max {2 p_{1}, 4}, \\ d > max {2 p_{1}, 4} and p_{a}^{*} ⩾ min {4, \frac{2 p_{1}}{p_{1} - 1}}, \\ d > max {2 p_{1}, 4} and p_{a}^{*} < min {4, \frac{2 p_{1}}{p_{1} - 1}} and m_{d} ⩾ min {4, \frac{2 p_{1}}{p_{1} - 1}} . \end{array}

Moreover, assume that

\begin{matrix} m_{d} < 2^{*} \end{matrix}

and, in the case of

d > 4

, let us further assume that

\begin{matrix} p_{a}^{*} ⩾ max {\frac{2 d p_{1}}{p_{1} (d - 2) + 8 - 2 d}, \frac{2 d p_{1} (p_{1} - 1)}{2 d (p_{1} - 1) + (d - 2) p_{1} (p_{1} - 3)}} . \end{matrix}

Then there exists, at least, a weak solution

u \in L^{\infty} (0, T; H \cap V_{2}) \cap L^{\vec{p}} (0, T; V_{\vec{p}}) \cap L^{\vec{m}} (Q_{T})

to the problem ( 1.3 )–( 1.6 ) in the sense of Definition 2.1 .

The following local existence result has been established in [4, Theorem 7.2] just in the case of $γ_{d} > 0$ .

Theorem 2.2 (Local existence).

Let Ω be a bounded domain in $R^{d}$ , $d ⩾ 2$ , with a Lipschitz-continuous boundary $\partial Ω$ . Assume that $k_{i} = k > 0$ and $q_{i} = q = 2$ for all $i \in {1, \dots, d}$ , and $u_{0} \in V_{2}$ , $f \in L^{{\vec{p}}^{'}} (Q_{T})$ or $f \in L^{2} (Q_{T})$ . If $γ_{d} > 0$ , assume that the following conditions are both fulfilled, $\begin{array}{l} \overline{q} < m_{d} if f \in L^{{\vec{p}}^{'}} (Q_{T}), or (2 ⩽ m_{d} \land \overline{q} < m_{d}) if f \in L^{2} (Q_{T}), \\ m_{d} ⩽ q_{a}^{*} or m_{d} < ϖ, ϖ : = \frac{p_{a}^{*} - 2}{p_{a}^{*}} \overline{p} + 2 . \end{array}$ Assume also that $\begin{matrix} m_{d} ⩽ \frac{2 d}{d - 2} if d > 2, or m_{d} : 1 < m_{d} < \infty if d = 2, \end{matrix}$ and that one and only one of the following conditions is verified, $\begin{array}{l} d > max {2, \overline{p}} and 4 ⩽ max {\frac{2 d}{d - 2}, p_{a}^{*}} = max {\frac{2 d}{d - 2}, \frac{d \overline{p}}{d - \overline{p}}, p_{d}}, \\ d ⩽ max {2, \overline{p}} . \end{array}$ Then there exists $t^{*} \in (0, T)$ , depending on the problem data, such that the problem ( 1.3 )–( 1.6 ) has, at least, a weak solution in the cylinder $Q_{t *}$ and in the sense of Definition 2.1 .

3. Energy inequality

In this section, we consider the initial and boundary-value problem (1.3)–(1.6) in the case of $\begin{matrix} (3.1) & γ_{i} ⩽ 0 \forall i \in {1, \dots, d}, \end{matrix}$ and we assume that one of the alternatives holds, $\begin{array}{l} (3.2) & f \in L^{{\vec{p}}^{'}} (Q_{T}), \\ (3.3) & f \in L^{{\vec{m}}^{'}} (Q_{T}) . \end{array}$ We assume that also holds $\begin{matrix} (3.4) & u_{0} \in H \cap V_{\vec{q}} . \end{matrix}$ To simplify the writing, let us fix the notation for the different energies associated with the problem (1.3)–(1.6), $\begin{array}{l} (3.5) & Φ (t) : = \frac{1}{2} {‖ u (t) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}}, \\ (3.6) & Ψ (t) : = \sum_{i = 1}^{d} (ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} - γ_{i} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}}) . \end{array}$ Note that, due to assumption (3.1), $Ψ (t) ⩾ 0$ for all $t \in [0, T]$ . Functions $Φ (t)$ and $Ψ (t)$ will be used, not only in the present section, but throughout the rest of the paper.

Theorem 3.1.
Let u be a weak solution to the problem ( 1.3 )–( 1.6 ) in the sense of Definition 2.1 and assume that ( 3.4 ) is verified together with the alternative (1) of Theorem 2.1 . In addition, assume that one of the conditions ( 3.2 ) or ( 3.3 ) holds. Then there exists an independent of t and positive constant C such that $\begin{matrix} (3.7) & \frac{d}{d t} Φ (t) + Ψ (t) ⩽ C \sum_{i = 1}^{d} {‖ f (t) ‖}_{L^{s_{i}} (Ω)}^{s_{i}} \forall t \in [0, T], \end{matrix}$ where $s_{i} = p_{i}^{'}$ and $C = C (Ω, d, p_{i}, ν_{i})$ if ( 3.2 ) holds, or $s_{i} = m_{i}^{'}$ and $C = C (m_{i}, γ_{i})$ if it is ( 3.3 ) holding.
Proof.
For the smallest integer $s > 1 + \frac{d}{2}$ , we define $V^{s} : = closure of V in the norm of W^{s, 2} (Ω)$ . Let ${v_{k}}_{k \in N}$ be an orthogonal family in $V^{s}$ and orthonormal in H. Given $n \in N$ , we consider the n-dimensional space $V^{n}$ spanned by $v_{1}$ , …, $v_{n}$ . For each $n \in N$ , we search for a Galerkin approximation $u^{n} (t)$ of (2.17) in the form $\begin{matrix} u^{n} (t) = \sum_{k = 1}^{n} c_{k}^{n} (t) v_{k}, v_{k} \in V^{n} . \end{matrix}$ This function is found by solving the following system of n nonlinear ordinary differential equations, with respect to the n unknowns $c_{1}^{n}, \dots, c_{n}^{n}$ , obtained from (2.17): $\begin{matrix} (3.8) & \begin{matrix} [b] & \int_{Ω} \frac{\partial u^{n} (t)}{\partial t} \cdot v_{k} d x + \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u^{n} (t) |}^{q_{i} - 2} \frac{\partial (D_{i} u^{n} (t))}{\partial t} \cdot D_{i} v_{k} d x \\ + \sum_{i = 1}^{d} ν_{i} \int_{Ω} {| D_{i} u^{n} (t) |}^{p_{i} - 2} D_{i} u^{n} (t) \cdot D_{i} v_{k} d x + \int_{Ω} [(u^{n} (t) \cdot \nabla) u^{n} (t)] \cdot v_{k} d x \\ = \int_{Ω} f (t) \cdot v_{k} d x + \sum_{i = 1}^{d} γ_{i} \int_{Ω} {| u^{n} (t) |}^{m_{i} - 2} u^{n} (t) \cdot v_{k} d x \end{matrix} \end{matrix}$ for $k \in {1, \dots, n}$ , with $\begin{array}{l} (3.9) & u^{n} (0) = u_{0}^{n} \\ (3.10) & D_{i} u^{n} (0) = u_{1, i}^{n} \end{array}$ and where $u_{0}^{n}, u_{1, i}^{n} \in V^{n}$ are chosen in such a way that $\begin{array}{l} (3.11) & u_{0}^{n} ⟶ u_{0} strongly in H, as n \to \infty, \\ (3.12) & u_{1, i}^{n} ⟶ D_{i} u_{0} strongly in L^{q_{i}} (Ω), as n \to \infty . \end{array}$ In the case of the alternative (1) of Theorem 2.1 holds, it was proven in [4, Lemma 5.1], under the assumptions (3.9)–(3.10) and (3.11)–(3.12), the existence of a n-upple $c^{n} (t) \equiv (c_{1}^{n} (t), \dots, c_{n}^{n} (t))$ such that $c^{n} (t)$ is a solution to the system (3.8)–(3.10) in the interval $[0, T]$ and such that $\begin{matrix} (3.13) & \begin{matrix} \frac{d}{d t} (\frac{1}{2} {‖ u^{n} (t) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u^{n} (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}}) + \sum_{i = 1}^{d} ν_{i} {‖ D_{i} u^{n} (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} \\ = \int_{Ω} f (t) \cdot u^{n} (t) d x + \sum_{i = 1}^{d} γ_{i} {‖ u^{n} (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} \forall t \in [0, T] . \end{matrix} \end{matrix}$ Moreover, in [4, Lemma 5.1], it was proved that the Galerkin approximations $u^{n} (t)$ satisfy $\begin{matrix} (3.14) & \begin{matrix} sup_{s \in [0, T]} {‖ u^{n} (s) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} (\frac{κ_{i}}{q_{i}} sup_{s \in [0, T]} {‖ D_{i} u^{n} (s) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} + ν_{i} \int_{0}^{T} {‖ D_{i} u^{n} (s) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} d s) \\ + | γ_{d} | {‖ u^{n} ‖}_{L^{m_{d}} (Q_{T})}^{m_{d}} ⩽ C, \end{matrix} \end{matrix}$ for some positive constant C not depending on n. Thus, in view of (3.14) and by means of separability and reflexivity, there exists a subsequence (still denoted by) $u^{n}$ and there exists a function u such that $\begin{array}{l} (3.15) & u^{n} ⟶ u weakly- * in L^{\infty} (0, T; H \cap V_{\vec{q}}), as n \to \infty, \\ (3.16) & u^{n} ⟶ u weakly in L^{\vec{p}} (0, T; V_{\vec{p}}), as n \to \infty, \\ (3.17) & u^{n} ⟶ u weakly in L^{\vec{m}} (Q_{T}), as n \to \infty . \end{array}$ From (3.15) and (3.16), it follows that $\begin{array}{l} (3.18) & D_{i} u^{n} ⟶ D_{i} u weakly- * in L^{\infty} (0, T; L^{q_{i}} (Q_{T})), as n \to \infty, \\ (3.19) & D_{i} u^{n} ⟶ D_{i} u weakly in L^{p_{i}} (Q_{T}), as n \to \infty \end{array}$ for all $i \in {1, \dots, d}$ .

Now, we integrate (3.13) between $t_{1}$ and $t_{2}$ , with $t_{1}, t_{2} \in [0, T]$ and $t_{1} < t_{2}$ , and using assumption (3.1), we obtain $\begin{matrix} (3.20) & \begin{matrix} {[\frac{1}{2} {‖ u^{n} (t) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u^{n} (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}}]}_{t = t_{0}}^{t = t_{1}} \\ + \sum_{i = 1}^{d} \int_{t_{0}}^{t_{1}} (ν_{i} {‖ D_{i} u^{n} (s) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} - γ_{i} {‖ u^{n} (s) ‖}_{L^{m_{i}} (Ω)}^{m_{i}}) d s = \int_{t_{0}}^{t_{1}} \int_{Ω} f \cdot u^{n} d x d s . \end{matrix} \end{matrix}$ Using the convergence results (3.15)–(3.19) and a classical property of weak limits, together with the notations (3.5)–(3.6), we obtain from (3.20), $\begin{matrix} {[Φ (t)]}_{t = t_{0}}^{t = t_{1}} + Ψ (t) ⩽ \int_{t_{0}}^{t_{1}} \int_{Ω} f \cdot u d x d s . \end{matrix}$ Thus, we can write for every $t, t + Δ t \in [0, T]$ , with $Δ t > 0$ , $\begin{matrix} (3.21) & \frac{1}{| Δ t |} {[Φ (s)]}_{s = t}^{s = t + Δ t} ⩽ - \frac{1}{| Δ t |} \int_{t}^{t + Δ t} Ψ (s) d s + \frac{1}{| Δ t |} \int_{t}^{t + Δ t} \int_{Ω} f \cdot u d x d s . \end{matrix}$ From (3.16)–(3.17), we infer that $u \in L^{\vec{p}} (0, T; V_{\vec{p}}) \cap L^{\vec{m}} (Q_{T})$ , which, together with one of the assumptions (3.2) or (3.3), shows that $\begin{matrix} ‖ D_{i} u ‖_{L^{p_{i}} (Ω)}^{p_{i}}, ‖ u ‖_{L^{m_{i}} (Ω)}^{m_{i}} \in L^{1} [0, T] \forall i \in {1, \dots, d}, and \int_{Ω} f \cdot u d x \in L^{1} [0, T] . \end{matrix}$ In consequence, by Lebesgue’s differentiation theorem, every term on the right-hand side of (3.21) has a limit, for a.e. $t \in [0, T]$ , as $Δ t \to 0$ . This in turn yields the existence of a limit of the left-hand side of this inequality, for a.e. $t \in [0, T]$ and as $Δ t \to 0$ . Whence we can write for all $t \in [0, T]$ $\begin{matrix} (3.22) & \frac{d}{d t} Φ (t) + Ψ (t) ⩽ \int_{Ω} | f (t) \cdot u (t) | d x . \end{matrix}$ If (3.2) holds, we can use Holder and Young inequalities together with Troisi’s inequality (2.9) to estimate the right-hand side of (3.22) as follows, $\begin{matrix} (3.23) & \int_{Ω} | f (t) \cdot u (t) | d x ⩽ C \sum_{i = 1}^{d} {‖ f (t) ‖}_{L^{p_{i}^{'}} (Ω)}^{p_{i}^{'}} + \sum_{i = 1}^{d} \frac{ν_{i}}{2} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}}, \end{matrix}$ for some positive constant $C = C (Ω, d, p_{i}, ν_{i})$ . Alternatively, if (3.3) holds, we use Holder and Young inequalities so that $\begin{matrix} (3.24) & \int_{Ω} | f (t) \cdot u (t) | d x ⩽ C \sum_{i = 1}^{d} {‖ f (t) ‖}_{L^{m_{i}^{'}} (Ω)}^{m_{i}^{'}} + \sum_{i = 1}^{d} \frac{| γ_{i} |}{2} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}}, \end{matrix}$ for some positive constant $C = C (m_{i}, γ_{i})$ . Using (3.23) or (3.24), and observing the notations (3.5)–(3.6), we obtain (3.7). □

4. Large time behavior

Let u be a weak solution to the problem (1.3)–(1.6) in the conditions of Theorem 3.1 and let $Φ (t)$ be the energy function defined by (3.5). In this section, we are interested in the solutions to the problem (1.3)–(1.6) with a finite energy $Φ (t)$ . Therefore, we may assume that $\begin{matrix} (4.1) & Φ (t) ⩽ K for all t \in [0, T] for some constant K > 0 . \end{matrix}$ Normalizing (4.1), we obtain $\begin{matrix} (4.2) & E (t) < 1 for all t \in [0, T], where E (t) : = \frac{Φ (t)}{K} . \end{matrix}$ It follows from (3.5) and (4.2) that $\begin{matrix} (4.3) & \frac{1}{K} \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} < E (t) < 1 for all t \in [0, T] . \end{matrix}$

Next, we shall consider the exponential and polynomial time-decay properties of the solutions. We introduce the notation $\begin{matrix} (4.4) & μ : = \frac{2}{d} \sum_{i = 1}^{d} \frac{1}{q_{i}} . \end{matrix}$

4.1. Exponential time-decay

Theorem 4.1.
Let u be a weak solution to the problem ( 1.3 )–( 1.6 ) in the conditions of Theorem 3.1 . Assume that one of the following alternatives holds:
$μ ⩾ 1$ , $q_{i} = p_{i}$ for all $i \in {1, \dots, d}$ and $\begin{matrix} (4.5) & q_{a}^{} ⩾ 2; \end{matrix}$

$m_{i} = 2$ and* $q_{i} = p_{i}$ for all $i \in {1, \dots, d}$ .
Assume also that $\begin{matrix} \int_{0}^{\infty} {‖ f (s) ‖}_{L_{s_{i}} (Ω)}^{s_{i}} d s < \infty \forall i \in {1, \dots, d}, \end{matrix}$ where $s_{i} = p_{i}^{'}$ if ( 3.2 ) holds, or $s_{i} = m_{i}^{'}$ if it is ( 3.3 ) holding. Then there exist two positive constants $C_{1}$ and $C_{2}$ such that $\begin{matrix} (4.6) & Φ (t) ⩽ e^{- C_{1} t} (Φ (0) + C_{2} \sum_{i = 1}^{d} \int_{0}^{t} e^{C_{1} s} {‖ f (s) ‖}_{L_{s_{i}} (Ω)}^{s_{i}} d s) . \end{matrix}$
Proof.
(1) Let $μ ⩾ 1$ and assume that $q_{i} = p_{i}$ for all $i \in {1, \dots, d}$ . Then, in view of (4.5), we can use Troisi’s inequality (2.9), together with (4.3) and with the facts that $γ_{i} ⩽ 0$ for all $i \in {1, \dots, d}$ and $μ ⩾ 1$ (see assumptions (3.1) and (4.5)), so that $\begin{matrix} (4.7) & \begin{matrix} Φ (t) & ⩽ \frac{C_{1}^{'}}{2} {(\sum_{i = 1}^{d} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}})}^{μ} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} \\ ⩽ \frac{C_{1}^{'}}{2} {[(max_{i \in {1, \dots, d}} {\frac{q_{i}}{κ_{i}}} K) \frac{1}{K} \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}}]}^{μ} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} \\ ⩽ [\frac{C_{1}^{'}}{2} {(max_{i \in {1, \dots, d}} {\frac{q_{i}}{κ_{i}}} K)}^{μ} + K] (\frac{1}{K} \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}}) \\ ⩽ [\frac{C_{1}^{'}}{2} {(max_{i \in {1, \dots, d}} {\frac{q_{i}}{κ_{i}}})}^{μ} K^{μ - 1} + 1] max_{i \in {1, \dots, d}} {\frac{κ_{i}}{q_{i} ν_{i}}} \sum_{i = 1}^{d} ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} \\ ⩽ C_{2}^{'} \sum_{i = 1}^{d} (ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} - γ_{i} ‖ u ‖_{L^{m_{i}} (Ω)}^{m_{i}}) = C_{2}^{'} Ψ (t), \end{matrix} \end{matrix}$ where μ is defined at (4.4), $C_{1}^{'}$ is the positive constant from the inequality (2.9) and $\begin{matrix} C_{2}^{'} = [\frac{C_{1}^{'}}{2} {(max_{i \in {1, \dots, d}} {\frac{q_{i}}{κ_{i}}})}^{μ} K^{μ - 1} + 1] max_{i \in {1, \dots, d}} {\frac{κ_{i}}{q_{i} ν_{i}}} . \end{matrix}$

(2) Now, we consider the case $m_{i} = 2$ and $q_{i} = p_{i}$ for all $i \in {1, \dots, d}$ . In this case, we have $\begin{matrix} (4.8) & \begin{matrix} Φ (t) & = \sum_{i = 1}^{d} (\frac{1}{2} ‖ u ‖_{L^{m_{i}} (Ω)}^{m_{i}} + \frac{κ_{i}}{p_{i}} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{q_{i}}) \\ ⩽ \frac{C_{3}^{″}}{2} \sum_{i = 1}^{d} (ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} + | γ_{i} | ‖ u ‖_{L^{m_{i}} (Ω)}^{m_{i}}) = C_{3}^{'} Ψ (t), \end{matrix} \end{matrix}$ where $C_{3}^{'} = \frac{C_{3}^{″}}{2}$ and $\begin{matrix} C_{3}^{″} = max {max_{i \in {1, \dots, d}} {\frac{1}{| γ_{i} |}}, 2 max_{i \in {1, \dots, d}} {\frac{κ_{i}}{q_{i} ν_{i}}}} . \end{matrix}$

Then, if we plug (4.7) or (4.8) into (3.7), we get, in any case, the following ordinary differential inequality $\begin{matrix} (4.9) & Φ^{'} (t) + C_{4}^{'} Φ (t) ⩽ C_{0} \sum_{i = 1}^{d} ‖ f ‖_{L^{s_{i}} (Ω)}^{s_{i}} for all t \in [0, T], \end{matrix}$ where $C_{4}^{'} = \frac{1}{C_{2}^{'}}$ in the case of the alternative (1), or $C_{4} = \frac{1}{C_{3}^{'}}$ in the case (2). After applying Gronwall’s lemma to (4.9), we obtain (4.6) with the constants $C_{1} = C_{4}^{'}$ and $C_{2} = C_{0}^{'}$ , where $C_{0}^{'}$ denotes here the constant from the right-hand side of the energy inequality (3.7). □
4.2. Polynomial time-decay

In order to state the main result of this subsection, let us define the following quantities $\begin{matrix} (4.10) & β : = \{\begin{matrix} β_{1} : = μ {min}_{i \in {1, \dots, d}} {\frac{q_{i}}{p_{i}}}, & if \sum_{i = 1}^{d} ν_{i} ‖ D_{i} u (t) ‖_{L^{p_{i}} (Ω)}^{p_{i}} ⩽ 1, \\ β_{2} : = μ {max}_{i \in {1, \dots, d}} {\frac{q_{i}}{p_{i}}}, & if \sum_{i = 1}^{d} ν_{i} ‖ D_{i} u (t) ‖_{L^{p_{i}} (Ω)}^{p_{i}} ⩾ 1, \end{matrix} \end{matrix}$ where μ is defined in (4.4) and $\begin{matrix} (4.11) & α : = \frac{1}{β}, \end{matrix}$ and where u is a weak solution to the problem (1.3)–(1.6) in the conditions of Theorem 3.1.

Theorem 4.2.
Let u be a weak solution to the problem ( 1.3 )–( 1.6 ) in the conditions of Theorem 3.1 and assume that ( 4.5 ) is verified.

In addition, consider the notation for α, defined above at ( 4.11 ), and assume that one of the following alternatives holds: $\begin{array}{l} (4.12) & μ < 1 and q_{i} ⩽ p_{i} \forall i \in {1, \dots, d}; \\ (4.13) & μ ⩽ 1 and q_{i} < p_{i} \forall i \in {1, \dots, d} . \end{array}$
If $f = 0$ a.e. in $Q_{T}$ , then there exist an independent of t positive constant $K_{0}$ such that $\begin{matrix} (4.14) & Φ (t) ⩽ K_{0} {(1 + t)}^{- \frac{1}{α - 1}} \forall t ⩾ 0 . \end{matrix}$

If $f \neq 0$ , but exist positive constants $C_{f}^{1}$ , …, $C_{f}^{d}$ and $σ_{1}$ , …, $σ_{d}$ such that $σ_{i} ⩾ \frac{α}{α - 1}$ for all $i \in {1, \dots, d}$ and $\begin{matrix} (4.15) & \int_{Ω} {| f (t) |}^{p_{i}^{'}} d x ⩽ C_{f}^{i} {(1 + t)}^{- σ_{i}} \forall t ⩾ 0, \forall i \in {1, \dots, d}, \end{matrix}$ or $\begin{matrix} (4.16) & \int_{Ω} {| f (t) |}^{m_{i}^{'}} d x ⩽ C_{f}^{i} {(1 + t)}^{- σ_{i}} \forall t ⩾ 0, \forall i \in {1, \dots, d}, \end{matrix}$ then there exist an independent of t positive constant $K_{f}$ such that $\begin{matrix} (4.17) & Φ (t) ⩽ K_{f} {(1 + t)}^{- \frac{α}{α - 1}} \forall t ⩾ 0 . \end{matrix}$

Remark 4.1.
Note that, by the alternatives (4.12) and (4.13), we have, in any case, $β < 1$ and therefore $\begin{matrix} (4.18) & α > 1 . \end{matrix}$
Proof.
Using (4.3) and (4.5), and reasoning as we did for (4.7), but using here the fact that $μ ⩽ 1$ instead of $μ ⩾ 1$ , we get the following chain of inequalities for $Φ (t)$ , $\begin{matrix} (4.19) & \begin{matrix} Φ (t) & ⩽ \frac{C_{1}^{'}}{2} {(\sum_{i = 1}^{d} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}})}^{μ} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} \\ ⩽ [\frac{C_{1}^{'}}{2} {(max_{i \in {1, \dots, d}} {\frac{q_{i}}{κ_{i}}} K)}^{μ} + K] {(\frac{1}{K} \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}})}^{μ} \\ ⩽ C_{2}^{'} {(\sum_{i = 1}^{d} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}})}^{μ} ⩽ C_{3}^{'} {(\sum_{i = 1}^{d} {(ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}})}^{\frac{q_{i}}{p_{i}}})}^{μ} \\ ⩽ C_{4}^{'} {(\sum_{i = 1}^{d} ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}})}^{β}, \end{matrix} \end{matrix}$ where β is defined above at (4.10). In the derivation of (4.19) we have denoted by $C_{1}^{'}$ the positive constant from (2.9) and we have used the following notations, $\begin{array}{l} C_{2}^{'} = [\frac{C_{1}^{'}}{2} {(max_{i \in {1, \dots, d}} {\frac{q_{i}}{κ_{i}}})}^{μ} + K^{1 - μ}] {(max_{i \in {1, \dots, d}} {\frac{κ_{i}}{q_{i}}})}^{μ}, \\ C_{3}^{'} = C_{2}^{'} {(max_{i \in {1, \dots, d}} {| Ω |^{\frac{p_{i} - q_{i}}{q_{i}}} ν_{i}^{- \frac{q_{i}}{p_{i}}}})}^{μ}, C_{4}^{'} = C_{3}^{'} d^{μ} . \end{array}$ In the expression for $C_{3}^{'}$ , $| Ω |$ denotes the d-Lebesgue measure of the domain Ω.

From (4.19) and (4.10), and observing (3.6), we get $\begin{matrix} (4.20) & Φ (t) ⩽ C_{4}^{'} {(\sum_{i = 1}^{d} ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}})}^{β} ⩽ C_{4}^{'} {(Ψ (t))}^{β}, \end{matrix}$ where $\begin{matrix} {(\sum_{i = 1}^{d} ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}})}^{β} : = max_{γ \in {β_{1}, β_{2}}} {(\sum_{i = 1}^{d} ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}})}^{γ} \end{matrix}$ and $β_{1}$ and $β_{2}$ are defined above at (4.10). Plugging (4.20) into (3.7), we obtain the following nonlinear differential inequality $\begin{matrix} (4.21) & Φ^{'} (t) + C_{1} {(Φ (t))}^{α} ⩽ C_{2} \sum_{i = 1}^{d} ‖ f ‖_{L^{s_{i}} (Ω)}^{s_{i}} \forall t \in [0, T], \end{matrix}$ where $C_{1} = {(C_{4}^{'})}^{- \frac{1}{β}}$ , $C_{2}$ is the constant from the right-hand side of the energy inequality (3.7), and α is defined above at (4.11).

(1) If $f = 0$ a.e. in $Q_{T}$ , we can, in view of (4.18), apply Lemma 2.2 to the nonlinear differential inequality (4.21) so that $\begin{matrix} Φ (t) ⩽ {(Φ {(0)}^{1 - α} + C_{1} (α - 1) t)}^{- \frac{1}{α - 1}} \forall t ⩾ 0, \end{matrix}$ which in turn implies (4.14), with $\begin{matrix} K_{0} : = {[max {Φ {(0)}^{1 - α}, C_{1} (α - 1)}]}^{- \frac{1}{α - 1}}, \end{matrix}$ and where, due to assumption (3.4), $\begin{matrix} Φ (0) = \frac{1}{2} ‖ u_{0} ‖_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} ‖ D_{i} u_{0} ‖_{L^{q_{i}} (Ω)}^{q_{i}} < \infty \end{matrix}$

(2) Now we assume that $f \neq 0$ and satisfies to (4.15), or alternatively to (4.16). Using any of these conditions in the nonlinear differential inequality (4.21), we obtain $\begin{matrix} Φ^{'} (t) + C_{1} Φ^{α} (t) ⩽ C_{2} \sum_{i = 1}^{d} {C_{f}^{i} (1 + t)}^{- σ_{i}} ⩽ C_{f} {(1 + t)}^{- \frac{α}{α - 1}} \forall t ⩾ 0, \end{matrix}$ where $C_{f} = C_{2} {max}_{i \in {1, \dots, d}} {C_{f}^{i}}$ , and $C_{1}$ , $C_{2}$ are the positive constants given in (4.21).

Let us now look for solutions of the form $y (t) = \tilde{C} {(1 + t)}^{- \frac{1}{α - 1}}$ to the ordinary differential equation $\begin{matrix} (4.22) & y^{'} (t) + C_{1} y^{α} (t) = C_{} {(1 + t)}^{- \frac{α}{α - 1}}, \end{matrix}$ where $C_{}$ is a positive constant to be found below and $\tilde{C}$ is a positive solution to the equation $\begin{matrix} f (\tilde{C}) = 0, f (\tilde{C}) : = - \frac{1}{α - 1} \tilde{C} + C_{1} {\tilde{C}}^{α} - C_{} . \end{matrix}$

Since, $f (0) = - C_{}$ , $C_{} > 0$ and, due to (4.18), ${lim}_{\tilde{C} \to + \infty} f (\tilde{C}) = + \infty$ , by Bolzano’s theorem the equation $f (\tilde{C}) = 0$ has a root $\overline{C} > 0$ . On the other hand, $\overline{C}$ can be estimated from below as follows $\begin{matrix} (4.23) & \overline{C} = {[\frac{1}{C_{1}} (C_{} + \frac{1}{α - 1} \overline{C})]}^{\frac{1}{α}} > {(\frac{C_{}}{C_{1}})}^{\frac{1}{α}} \end{matrix}$ or $\begin{matrix} \overline{C} = {[\frac{1}{C_{1}} (C_{} + \frac{1}{α - 1} \overline{C})]}^{\frac{1}{α}} > {(\frac{\overline{C}}{(α - 1) C_{1}})}^{\frac{1}{α}} \Rightarrow \overline{C} > {(\frac{1}{(α - 1) C_{1}})}^{\frac{1}{α - 1}} . \end{matrix}$ Next, we introduce the new function $\begin{matrix} G (t) : = Φ (t) - y (t) \equiv Φ (t) - \overline{C} {(1 + t)}^{- \frac{α}{α - 1}}, \end{matrix}$ which in turn satisfies the following linear differential inequality $\begin{matrix} (4.24) & G^{'} (t) + C_{3} G (t) ⩽ C_{4} {(1 + t)}^{- \frac{α}{α - 1}}, \end{matrix}$ where $\begin{matrix} C_{3} : = C_{1} α \int_{0}^{1} {[λ Φ (t) + (1 - λ) y (t)]}^{α - 1} d λ ⩾ 0 and C_{4} : = C_{F} - C_{} . \end{matrix}$ Now, we choose the constant $C_{}$ stated at (4.22) in such a way that $C_{4} ⩽ 0$ , i.e., $C_{} ⩾ C_{F}$ . Using this information, the linear differential inequality (4.24) becomes homogeneous. Solving the resulting homogeneous differential inequality, we obtain $\begin{matrix} (4.25) & G (t) ⩽ G (0) e^{- \int_{0}^{t} C_{3} (s) d s} ⩽ G (0) = Φ (0) - \overline{C} . \end{matrix}$ Imposing also that $\begin{matrix} G (0) ⩽ 0 \Leftrightarrow \overline{C} ⩾ Φ (0) = \frac{1}{2} ‖ u_{0} ‖_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} ‖ D_{i} u_{0} ‖_{L^{q_{i}} (Ω)}^{q_{i}}, \end{matrix}$ it follows, from (4.25), that $\begin{matrix} G (t) ⩽ 0 \forall t ⩾ 0 . \end{matrix}$ As a consequence, by the estimate (4.23), we can see that a choice of $C_{} = max {C_{F}, C_{1} Φ^{α} (0)}$ proves that $\begin{matrix} 0 ⩽ Φ (t) ⩽ \overline{C} {(1 + t)}^{- \frac{α}{α - 1}}, \end{matrix}$ and, finally, the decay property (4.17) follows by taking $K_{f} = \overline{C}$ . □

5. Blow-up without the convective term

In this section, we consider the problem (1.3)–(1.6) in the case of $f = 0$ and without the convective term, i.e. we consider the following initial and boundary value problem $\begin{array}{l} (5.1) & \partial_{t} u + \nabla π = \sum_{i = 1}^{d} D_{i} (κ_{i} | D_{i} u |^{q_{i} - 2} \partial_{t} (D_{i} u) + ν_{i} | D_{i} u |^{p_{i} - 2} D_{i} u) + \sum_{i = 1}^{d} γ_{i} | u |^{m_{i} - 2} u in Q_{T}, \\ (5.2) & div u = 0 in Q_{T}, \\ (5.3) & u = u_{0} in Ω for t = 0, \\ (5.4) & u = 0 on Γ_{T} . \end{array}$

Before we state the next result, we recall that, from what we have assumed in (2.1)–(2.3), $\begin{matrix} q_{d} = max_{i \in {1, \dots, d}} q_{i} and m_{1} = min_{i \in {1, \dots, d}} m_{i} . \end{matrix}$ Recall also the notations (3.5) and (3.6) for the functions $Φ (t)$ and $Ψ (t)$ .

Theorem 5.1.
Let $κ_{i} ⩾ 0$ , $ν_{i} > 0$ and $γ_{i} > 0$ for all $i \in {1, \dots, d}$ , and assume that $\begin{array}{l} (5.5) & m_{i} ⩾ p_{i} \forall i \in {1, \dots, d}, \\ (5.6) & m_{1} > max {2, q_{d}} \end{array}$ and that $\begin{matrix} (5.7) & \sum_{i = 1}^{d} (\frac{γ_{i}}{m_{i}} ‖ u_{0} ‖_{L^{m_{i}} (Ω)}^{m_{i}} - \frac{ν_{i}}{p_{i}} ‖ D_{i} u_{0} ‖_{L^{p_{i}} (Ω)}^{p_{i}}) ⩾ 0 . \end{matrix}$ Then there exists a finite time $T_{max} < \infty$ such that the strong solutions to the problem ( 5.1 )–( 5.4 ) blow-up in the following sense $\begin{matrix} Φ (t) = \frac{1}{2} {‖ u (t) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} ⟶ \infty, as t \to T_{max} . \end{matrix}$
Remark 5.1.
If $κ_{i} = 0$ for all $i \in {1, \dots, d}$ , then the system (5.1)–(5.4) becomes the Stokes problem with anisotropic diffusion and damping. In this case, the blow-up property still holds.
Proof.
The proof of this theorem is based on the method worked out in [10]. Let us introduce the function $\begin{matrix} (5.8) & Υ (t) : = \int_{0}^{t} Φ (s) d s, t > 0, \end{matrix}$ where $Φ (s)$ stands for the energy function defined at (3.5). For every nontrivial solution to the problem (5.1)–(5.4), we have $\begin{matrix} (5.9) & Υ^{'} (t) = Φ (t) ⩾ 0, \forall t > 0 . \end{matrix}$ Multiplying, formally, the equation (5.1), first by u and then by $u_{t}$ , and integrating the resulting equations over Ω, we get, respectively, the following energy relations, $\begin{array}{l} (5.10) & \frac{d}{d t} Φ (t) = \sum_{i = 1}^{d} (γ_{i} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} - ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}}), \\ {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u |}^{q_{i} - 2} {| D_{i} u_{t} (t) |}^{2} d x \\ (5.11) & = \frac{d}{d t} \sum_{i = 1}^{d} (\frac{γ_{i}}{m_{i}} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} - \frac{ν_{i}}{p_{i}} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}}) . \end{array}$ Plugging (5.9) into (5.10), we get $\begin{matrix} (5.12) & Υ^{″} (t) = \sum_{i = 1}^{d} (γ_{i} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} - ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}}) . \end{matrix}$ Integrating (5.11) from 0 to $t \in (0, T)$ , and then, using assumption (5.7), we obtain $\begin{matrix} (5.13) & \begin{matrix} \int_{0}^{t} ({‖ u_{s} (s) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (s) |}^{q_{i} - 2} {| D_{i} u_{s} (s) |}^{2} d x) d s \\ ⩽ \sum_{i = 1}^{d} (\frac{γ_{i}}{m_{i}} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} - \frac{ν_{i}}{p_{i}} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}}) . \end{matrix} \end{matrix}$ Using the assumption (5.5) together with (2.3), and observing the identity (5.12), we can estimate the right-hand side of (5.13) as follows, $\begin{matrix} \sum_{i = 1}^{d} (\frac{γ_{i}}{m_{i}} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} - \frac{ν_{i}}{p_{i}} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}}) ⩽ \frac{1}{m_{1}} Υ^{″} (t) . \end{matrix}$ Combining this inequality with (5.13), we obtain $\begin{matrix} (5.14) & \int_{0}^{t} ({‖ u_{s} (s) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (s) |}^{q_{i} - 2} {| D_{i} u_{s} (s) |}^{2} d x) d s ⩽ \frac{1}{m_{1}} Υ^{″} (t) . \end{matrix}$ Let us denote by $t^{}$ the time of existence of the solution u, $\begin{matrix} t^{} = sup {t > 0 : sup_{t \in (0, T)} Φ (t) < \infty \forall t < t^{}}, \end{matrix}$ where $Φ (t)$ is the energy function defined at (3.5). By virtue of (5.8), (5.9) and (5.14), we can infer that, for every nonstationary solution u the problem (5.1)–(5.4), $\begin{matrix} \exists t^{'} > 0 : Υ (t) > 0, Υ^{'} (t) > 0, Υ^{″} (t) > 0 \forall t > t^{'} . \end{matrix}$ In order to obtain a contradiction, let us assume the blow-up does not occur in a finite time, i.e., the nonstationary solution u exists for all time $t > 0$ and so $t^{} = \infty$ .

Using (5.9) and (5.14) together with the Hölder inequality, we get the following chain of inequalities $\begin{matrix} (5.15) & \begin{matrix} {[Υ^{'} (t) - Υ^{'} (t^{'})]}^{2} \\ = {[\int_{t^{'}}^{t} Υ^{″} (s) d s]}^{2} \\ = {[\int_{t^{'}}^{t} (\int_{Ω} u (s) \cdot u_{s} (s) d x + \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (s) |}^{q_{i} - 2} D_{i} (u (s)) D_{i} (u_{s} (s)) d x) d s]}^{2} \\ ⩽ [{(\int_{t^{'}}^{t} {‖ u (s) ‖}_{L^{2} (Ω)}^{2} d s)}^{\frac{1}{2}} {(\int_{t^{'}}^{t} {‖ u_{s} (s) ‖}_{2}^{2} d s)}^{\frac{1}{2}} \\ + \sum_{i = 1}^{d} {(\int_{t^{'}}^{t} κ_{i} {‖ D_{i} u (s) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} d s)}^{\frac{1}{2}} {(\int_{t^{'}}^{t} κ_{i} \int_{Ω} {| D_{i} u (s) |}^{q_{i} - 2} {| D_{i} (u_{s} (s)) |}^{2} d x d s)}^{\frac{1}{2}}]^{2} \\ ⩽ [\int_{t^{'}}^{t} ({‖ u (s) ‖}_{L^{2} (Ω)}^{2} d s + \sum_{i = 1}^{d} κ_{i} {‖ D_{i} u (s) ‖}_{L^{q_{i}} (Ω)}^{q_{i}}) d s] \\ \times [\int_{t^{'}}^{t} ({‖ u_{s} (s) ‖}_{L^{2} (Ω)}^{2} d s + \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (s) |}^{q_{i} - 2} {| D_{i} (u_{s} (s)) |}^{2} d x) d s] \\ ⩽ \frac{max {2, q_{d}}}{m_{1}} Υ (t) Υ^{″} (t) \forall t > t^{'} > 0 . \end{matrix} \end{matrix}$ Since $Υ (t) > 0$ , $Υ^{'} (t) > 0$ and $Υ^{″} (t) > 0$ for all $t ⩾ t^{'}$ , it follows that $Υ^{'} (t) ↗ \infty$ , as $t \to \infty$ . From here, and by virtue of assumption (5.6), we can infer that, for every σ such that $1 < σ < \frac{m_{1}}{max {2, q_{d}}}$ , one has $\begin{matrix} 1 - \sqrt{\frac{max {2, q_{d}} σ}{m_{1}}} ⩾ \frac{Υ^{'} (t^{'})}{Υ^{'} (t)} ⟶ 0, as t \to \infty . \end{matrix}$ Thus, from (5.15), it follows that for every fixed $σ \in (1, \frac{m_{1}}{max {2, q_{d}}})$ there exists a moment $t_{0} > t^{'}$ such that $\begin{matrix} {(Υ^{'} (t) - Υ^{'} (t^{'}))}^{2} ⩾ \frac{max {2, q_{d}} σ}{m_{1}} {(Υ^{'} (t))}^{2} \forall t > t_{0}, Υ (t_{0}) > 0 . \end{matrix}$ Using (5.15) together with last inequality, we get $\begin{matrix} \frac{max {2, q_{d}} σ}{m_{1}} {(Υ^{'} (t))}^{2} ⩽ {(Υ^{'} (t) - Υ^{'} (t^{'}))}^{2} ⩽ \frac{max {2, q_{d}}}{m_{1}} Υ^{″} (t) Υ (t) \forall t > t_{0} . \end{matrix}$ Hence $\begin{matrix} (5.16) & \begin{matrix} σ \frac{Υ^{'} (t)}{Υ (t)} ⩽ \frac{Υ^{″} (t)}{Υ^{'} (t)} \\ \Rightarrow {(ln Υ^{σ} (t))}^{'} ⩽ {(ln Υ^{'} (t))}^{'} \\ \Rightarrow (\frac{Υ^{'} (t_{0})}{Υ^{σ} (t_{0})}) Υ^{σ} (t) ⩽ Υ^{'} (t) \forall t > t_{0} . \end{matrix} \end{matrix}$ Integrating (5.16) between $t_{0}$ and t, leads us to the inequality $\begin{matrix} Υ^{σ - 1} (t) ⩾ \frac{Υ^{σ - 1} (t_{0})}{1 - (t - t_{0}) (σ - 1) \frac{Υ^{'} (t_{0})}{Υ (t_{0})}} \to \infty, as t \to T_{max} = t_{0} + \frac{Υ (t_{0})}{(σ - 1) Υ^{'} (t_{0})} . \end{matrix}$ As a consequence, we have $\begin{matrix} \infty > T \cdot sup_{t \in (0, T)} Φ (t) ⩾ \int_{0}^{t} Φ (s) d s \equiv Υ (t) ⟶ \infty, as t \to T_{max}, \end{matrix}$ which contradicts our assumption that $t^{*} = \infty$ . □

6. Blow-up with anisotropic relaxation, diffusion and damping

In this section we consider the problem (1.3)–(1.6) with $f = 0$ and $γ_{i} > 0$ for all $i \in {1, \dots, d}$ . Recall that, from what we have assumed in (2.1)–(2.3) $\begin{matrix} q_{d} = max_{i \in {1, \dots, d}} q_{i} and m_{d} = max_{i \in {1, \dots, d}} m_{i} . \end{matrix}$ Recall also here the notations (3.5) and (3.6) for the functions $Φ (t)$ and $Ψ (t)$ . In particular, taking $t = 0$ in (3.5) and observing (1.5), we have $\begin{matrix} Φ (0) = \frac{1}{2} {‖ u_{0} ‖}_{2, Ω}^{2} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} ‖ D_{i} u_{0} ‖_{L^{q_{i}} (Ω)}^{q_{i}} . \end{matrix}$

Theorem 6.1.
Let be $κ_{i}$ , $ν_{i}$ , $γ_{i} > 0$ for all $i \in {1, \dots, d}$ and assume that $\begin{array}{l} (6.1) & \frac{q_{i}}{2} ⩽ p_{i} ⩽ q_{i}, q_{i} > 2 \forall i \in {1, \dots, d}, \\ (6.2) & m_{d} q_{i} - 2 q_{i} - 2 m_{d} ⩾ 0, \forall i \in {1, \dots, d}, \\ (6.3) & m_{d} ⩾ 2 (m_{i} - 1) \forall i \in {1, \dots, d - 1}, \\ (6.4) & m_{d} > q_{d} . \end{array}$ Assume, in addition, that $\begin{matrix} (6.5) & \sum_{i = 1}^{d} γ_{i} ‖ u_{0} ‖_{L^{m_{i}} (Ω)}^{m_{i}} ⩾ \sum_{i = 1}^{d} ν_{i} ‖ D_{i} u_{0} ‖_{L^{p_{i}} (Ω)}^{p_{i}} + C_{1} Φ (0) + C_{2}, \end{matrix}$ where $C_{1}$ and $C_{2}$ are positive constants, defined below at ( 6.30 ).

Then there exists a finite time $T_{max} < \infty$ such that the strong solutions to the problem ( 1.3 )–( 1.6 ) blow-up in the following sense $\begin{matrix} (6.6) & Φ (t) = \frac{1}{2} {‖ u (t) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} \to \infty, as t \to T_{max} . \end{matrix}$
Remark 6.1.
By Lemma 2.3 one can takes the upper bound of the interval $(0, T)$ for $T_{max}$ (see (6.33) below).
Proof.
The proof of this theorem is based on the method developed in [9]. Let us introduce the functions $\begin{array}{l} (6.7) & I (t) : = Φ (t) + C_{0}, \\ (6.8) & J (t) : = {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (t) |}^{q_{i} - 2} {| D_{i} u_{t} (t) |}^{2} d x, t > 0, \end{array}$ where $Φ (t)$ is the energy function defined at (3.5) and $C_{0}$ is a nonnegative constant to be chosen later on.

Multiplying, formally, the equation (1.3) first by u and then by $u_{t}$ , and integrating the resulting equations over Ω, we get, respectively, the following energy relations, $\begin{matrix} (6.9) & \frac{d}{d t} Φ (t) + \sum_{i = 1}^{d} ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} = \sum_{i = 1}^{d} γ_{i} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} \end{matrix}$ and $\begin{matrix} (6.10) & \begin{matrix} {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (t) |}^{q_{i} - 2} {| D_{i} u_{t} (t) |}^{2} d x + \frac{d}{d t} \sum_{i = 1}^{d} \frac{ν_{i}}{p_{i}} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} \\ = \frac{d}{d t} \sum_{i = 1}^{d} \frac{γ_{i}}{m_{i}} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} - \int_{Ω} (u (t) \cdot \nabla) u (t) \cdot u_{t} (t) d x . \end{matrix} \end{matrix}$ Using the notations introduced above at (6.7)–(6.8), we can rewrite (6.9) as $\begin{matrix} (6.11) & I^{'} (t) + \sum_{i = 1}^{d} ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} = \sum_{i = 1}^{d - 1} γ_{i} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} + γ_{d} {‖ u (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}}, \end{matrix}$ and (6.10) as $\begin{matrix} (6.12) & \begin{matrix} J (t) + \frac{d}{d t} \sum_{i = 1}^{d} \frac{ν_{i}}{p_{i}} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} \\ = \frac{d}{d t} \sum_{i = 1}^{d - 1} \frac{γ_{i}}{m_{i}} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} + \frac{γ_{d}}{m_{d}} \frac{d}{d t} {‖ u (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}} - \int_{Ω} (u (t) \cdot \nabla) u (t) \cdot u_{t} (t) d x . \end{matrix} \end{matrix}$ Deriving (6.11) with respect to t, and then combining the resulting equation with (6.12), gives $\begin{array}{rcl} J (t) & = & \frac{1}{m_{d}} I^{″} (t) + \frac{d}{d t} \sum_{i = 1}^{d} ν_{i} \frac{(p_{i} - m_{d})}{m_{d} p_{i}} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}} + \frac{d}{d t} \sum_{i = 1}^{d - 1} γ_{i} \frac{(m_{d} - m_{i})}{m_{d} m_{i}} {‖ u (t) ‖}_{L^{m_{i}} (Ω)}^{m_{i}} \\ (6.13) & - \int_{Ω} (u (t) \cdot \nabla) u (t) \cdot u_{t} (t) d x : = \frac{1}{m_{d}} I^{″} (t) + J_{1} (t) + J_{2} (t) + J_{3} (t) . \end{array}$

On the other hand, observing the notation set forth at (3.5), it follows from (6.7) that $\begin{matrix} (6.14) & I^{'} (t) = \int_{Ω} u \cdot u_{t} d x + \sum_{i = 1}^{d} κ_{i} \int_{Ω} | D_{i} u |^{q_{i} - 2} D_{i} u (t) : D_{i} u_{t} (t) d x . \end{matrix}$ Arguing as we did for (5.15) and using the expression (6.14), we can estimate $| I^{'} (t) |^{2}$ as follows, $\begin{matrix} (6.15) & {| I^{'} (t) |}^{2} ⩽ max {2, q_{d}} I (t) J (t) \end{matrix}$ We estimate the term $J_{1} (t)$ on right-hand side of (6.13) by applying Cauchy’s inequality with ε. Next, for each $i \in {1, \dots, d}$ , we use Young’s inequality (2.13) with $ϵ = ε_{i}$ , to be defined later on, and $p = \frac{q_{i}}{2 p_{i} - q_{i}}$ , which is possible due to assumption (6.1)₁ $\begin{array}{rcl} | J_{1} (t) | & ⩽ & \sum_{i = 1}^{d} \frac{ν_{i} | p_{i} - m_{d} |}{m_{d}} \int_{Ω} {| D_{i} u (t) |}^{p_{i} - 1} | D_{i} u_{t} (t) | d x \\ ⩽ & \frac{ε}{2 m_{d}} \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (t) |}^{q_{i} - 2} {| D_{i} u_{t} (t) |}^{2} d x + \frac{1}{2 ε m_{d}} \sum_{i = 1}^{d} \frac{{(ν_{i} | p_{i} - m_{d} |)}^{2}}{κ_{i}} \int_{Ω} {| D_{i} u (t) |}^{2 p_{i} - q_{i}} d x \\ ⩽ & \frac{ε}{2 m_{d}} \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (t) |}^{q_{i} - 2} {| D_{i} u_{t} (t) |}^{2} d x \\ (6.16) & + \frac{1}{2 ε m_{d}} \sum_{i = 1}^{d} \frac{{(ν_{i} | p_{i} - m_{d} |)}^{2} (2 p_{i} - q_{i})}{κ_{i}^{2}} ε_{i}^{\frac{q_{i}}{2 p_{i} - q_{i}}} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} + J_{11}, \end{array}$ where $\begin{matrix} J_{11} : = \frac{| Ω |}{ε m_{d}} \sum_{i = 1}^{d} \frac{{(ν_{i} | p_{i} - m_{d} |)}^{2} (q_{i} - p_{i})}{κ_{i} q_{i} ε_{i}^{\frac{q_{i}}{2 (q_{i} - p_{i})}}} . \end{matrix}$

For the term $J_{2} (t)$ on the right-hand side of (6.13), we also first use Cauchy’s inequality with $ϵ = \frac{ε}{d - 1}$ , with $ε > 0$ to be defined later on. Then, for each $i \in {1, \dots, d}$ and in view of assumption (6.3), we can use Young’s inequality (2.13) with $ϵ = δ_{i}$ , to be chosen below, and with $p = \frac{m_{d}}{2 (m_{i} - 1)}$ . After all, we obtain $\begin{matrix} | J_{2} (t) | \\ ⩽ \sum_{i = 1}^{d} \frac{γ_{i} | m_{d} - m_{i} |}{m_{d}} \int_{Ω} {| u (t) |}^{m_{i} - 1} | u_{t} (t) | d x \\ ⩽ \frac{ε}{2 m_{d}} {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \frac{d - 1}{2 m_{d} ε} \sum_{i = 1}^{d - 1} (γ_{i}^{2} | m_{d} - m_{i} |^{2} \int_{Ω} {| u (t) |}^{2 (m_{i} - 1)} d x) \\ ⩽ \frac{ε}{2 m_{d}} {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} \\ + \frac{d - 1}{2 m_{d} ε} \sum_{i = 1}^{d - 1} (\frac{2 (m_{i} - 1) δ_{i}^{\frac{m_{d}}{2 (m_{i} - 1)}}}{m_{d}} {‖ u (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}} \\ + \frac{m_{d} - 2 (m_{i} - 1)}{m_{d} δ_{i}^{\frac{m_{d}}{m_{d} - 2 (m_{i} - 1)}}} | Ω | {((m_{d} - m_{i}) γ_{i})}^{\frac{2 m_{d}}{m_{d} - 2 (m_{i} - 1)}}) . \end{matrix}$ Now, for each $i \in {1, \dots, d}$ , we choose $δ_{i} = {(\frac{m_{d}}{2 (m_{i} - 1) {(d - 1)}^{2}})}^{\frac{2 (m_{i} - 1)}{m_{d}}}$ so that $\begin{matrix} (6.17) & | J_{2} (t) | ⩽ \frac{ε}{2 m_{d}} {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \frac{1}{2 m_{d} ε} {‖ u (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}} + J_{22}, \end{matrix}$ where $\begin{matrix} J_{22} : = \frac{| Ω |}{2 m_{d} ε} \sum_{i = 1}^{d - 1} \frac{m_{d} - 2 (m_{i} - 1)}{2 (d - 1) (m_{i} - 1)} {(\frac{2 (m_{i} - 1) {(γ_{i} (d - 1) | m_{d} - m_{i} |)}^{2}}{m_{d}})}^{\frac{m_{d}}{m_{d} - 2 (m_{i} - 1)}} . \end{matrix}$

For the term $J_{3} (t)$ , we proceed analogously, using, in this case, Cauchy’s inequality with $ϵ = \frac{ε}{2 m_{d}}$ , with $ε > 0$ to be defined later on, and next, for each $i \in {1, \dots, d}$ , we use Young’s inequality (2.13) with $ϵ = θ_{i}$ , with $θ_{i} > 0$ to be chosen later on, and with $p = \frac{m_{d}}{2}$ , which is possible due to assumption (6.2)₂. This way, we obtain $\begin{array}{rcl} | J_{3} (t) | & = & | - \sum_{i = 1}^{d} \int_{Ω} u_{i} (t) D_{i} u (t) \cdot u_{t} (t) d x | ⩽ \frac{ε}{2 m_{d}} {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \frac{d m_{d}}{2 ε} \sum_{i = 1}^{d} \int_{Ω} {| u_{i} (t) |}^{2} {| D_{i} u (t) |}^{2} d x \\ ⩽ & \frac{ε}{2 m_{d}} {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \frac{1}{ε} \sum_{i = 1}^{d} {‖ u_{i} (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}} + \frac{(m_{d} - 2) d^{\frac{m_{d}}{m_{d} - 2}}}{2 ε} \sum_{i = 1}^{d} \int_{Ω} {| D_{i} u (t) |}^{\frac{2 m_{d}}{m_{d} - 2}} d x \\ (6.18) & ⩽ & \frac{ε}{2 m_{d}} {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \frac{1}{ε} {‖ u (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}} + \frac{m_{d}}{ε} \sum_{i = 1}^{d} \frac{1}{q_{i}} θ_{i}^{\frac{q_{i} (m_{d} - 2)}{2 m_{d}}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} + J_{33}, \end{array}$ where $\begin{matrix} J_{33} : = \frac{| Ω |}{2 ε} \sum_{i = 1}^{d} \frac{m_{d} q_{i} - 2 q_{i} - 2 m_{d}}{q_{i}} {(\frac{d^{m_{d}}}{θ_{i}^{m_{d} - 2}})}^{\frac{q_{i}}{m q_{i} - 2 q_{i} - 2 m_{d}}} . \end{matrix}$

Plugging (6.16), (6.17) and (6.18) into (6.13), we obtain $\begin{matrix} (6.19) & \begin{matrix} (1 - \frac{ε}{m_{d}}) J (t) \\ ⩽ (1 - \frac{ε}{m_{d}}) {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + (1 - \frac{ε}{2 m_{d}}) \sum_{i = 1}^{d} κ_{i} \int_{Ω} {| D_{i} u (t) |}^{q_{i} - 2} {| D_{i} u_{t} (t) |}^{2} d x \\ ⩽ \frac{1}{m_{d}} I^{″} (t) + \frac{1 + 2 m_{d}}{2 m_{d} ε} {‖ u (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}} + \frac{A}{2 m_{d} ε} \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} + Λ, \end{matrix} \end{matrix}$ where $\begin{matrix} (6.20) & \begin{matrix} A : = max_{i \in {1, \dots, d}} {\frac{{(ν_{i} | p_{i} - m_{d} |)}^{2} (2 p_{i} - q_{i})}{κ_{i}^{2}} ε_{i}^{\frac{q_{i}}{2 p_{i} - q_{i}}} + \frac{2 m_{d}^{2}}{κ_{i}} θ_{i}^{\frac{q_{i} (m_{d} - 2)}{2 m_{d}}}}, \\ Λ : = J_{11} + J_{22} + J_{33} . \end{matrix} \end{matrix}$ Owe to assumption (6.1)₁, we can use, for each $i \in {1, \dots, d}$ , the continuous imbedding $L^{q_{i}} (Ω) ↪ L^{p_{i}} (Ω)$ , to get from (6.11) the following inequality $\begin{matrix} (6.21) & \begin{matrix} {‖ u (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}} ⩽ & \frac{1}{γ_{d}} (I^{'} (t) + \sum_{i = 1}^{d} ν_{i} {‖ D_{i} u (t) ‖}_{L^{p_{i}} (Ω)}^{p_{i}}) \\ ⩽ & \frac{1}{γ_{d}} I^{'} (t) + q_{d} \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} \\ + \frac{| Ω |}{γ_{d}} \sum_{i = 1}^{d} \frac{q_{i} - p_{i}}{q_{i}} ν_{i}^{\frac{q_{i}}{q_{i} - p_{i}}} {(\frac{p_{i}}{q_{d} γ_{d} κ_{i}})}^{\frac{p_{i}}{q_{i} - p_{i}}} . \end{matrix} \end{matrix}$ For each $i \in {1, \dots, d}$ , we now choose $ε_{i}$ and $θ_{i}$ , considered in the estimates (6.16) and (6.18), respectively, so that $\begin{matrix} (6.22) & ε_{i} = {(\frac{2 m_{d} q_{d} κ_{i}^{2}}{{(ν_{i} | p_{i} - m_{d} |)}^{2} (2 p_{i} - q_{i})})}^{\frac{2 p_{i} - q_{i}}{q_{i}}}, θ_{i} = {(2 q_{d} κ_{i})}^{\frac{2 m_{d}}{q_{i} (m_{d} - 2)}} . \end{matrix}$ Then plugging (6.21) into (6.19), one has $\begin{matrix} (6.23) & (1 - \frac{ε}{m_{d}}) J (t) ⩽ \frac{1}{m_{d}} I^{″} (t) + \frac{1 + 2 m_{d}}{2 m_{d} γ_{d} ε} I^{'} (t) + \frac{q_{d} {(1 + 2 m_{d})}^{2}}{2 m_{d} ε} \sum_{i = 1}^{d} \frac{κ_{i}}{q_{i}} {‖ D_{i} u (t) ‖}_{L^{q_{i}} (Ω)}^{q_{i}} + C_{0}^{'}, \end{matrix}$ where $\begin{matrix} (6.24) & C_{0}^{'} : = \frac{(1 + 2 m_{d}) | Ω |}{2 m_{d} ε γ_{d}} \sum_{i = 1}^{d} \frac{q_{i} - p_{i}}{q_{i}} ν_{i}^{\frac{q_{i}}{q_{i} - p_{i}}} {(\frac{p_{i}}{q_{d} γ_{d} κ_{i}})}^{\frac{p_{i}}{q_{i} - p_{i}}} + Λ, \end{matrix}$ where Λ is defined at (6.20). Gathering (6.7) and (6.23), we obtain $\begin{matrix} (6.25) & (1 - \frac{ε}{m_{d}}) J (t) ⩽ \frac{1}{m_{d}} I^{″} (t) + \frac{1 + 2 m_{d}}{2 m_{d} ε γ_{d}} I^{'} (t) + \frac{q_{d} {(1 + 2 m_{d})}^{2}}{2 m_{d} ε} (I (t) - C_{0}) + C_{0}^{'}, \end{matrix}$ where $C_{0}^{'}$ is defined above at (6.24) and $C_{0}$ is the constant from (6.7) that is chosen as satisfying the identity $\begin{matrix} (6.26) & C_{0} = \frac{2 m ε}{q {(1 + 2 m)}^{2}} C_{0}^{'} . \end{matrix}$

Then, combining (6.25)–(6.26) with (6.15), and using the assumptions (6.1)₂ and (6.4), we have $\begin{matrix} (6.27) & I^{″} (t) I (t) - (1 + α) {(I^{'} (t))}^{2} ⩾ - 2 B_{1} I^{'} (t) I (t) - B_{2} {(I (t))}^{2}, \end{matrix}$ where we have set $\begin{matrix} (6.28) & α : = \frac{m_{d} - ε}{q_{d}} - 1, B_{1} : = \frac{1 + 2 m_{d}}{4 ε γ_{d}} and B_{2} : = \frac{q_{d} {(1 + 2 m_{d})}^{2}}{2 ε} . \end{matrix}$

Observing again the assumption (6.4), we can choose $\begin{matrix} (6.29) & ε \in (0, m_{d} - q_{d}) \end{matrix}$ so that $α > 0$ . Thus, we can now apply Lemma 2.3, by taking there $Φ (t) = I (t)$ , with $I (t)$ defined by (6.7), and so that (6.27) matches with (2.10) for α, $B_{1}$ and $B_{2}$ defined at (6.28). In particular, first and third conditions of the assumption (2.11) are immediately satisfied in view of (3.5), (6.7) and (6.26), and of (6.28), respectively. With respect to the second condition of (2.11), we can see that, by using the notations (3.5), (6.7), (6.26) and (6.28) together with the identity (6.11), it is satisfied, because the inequality (6.5) is assumed to be verified for $\begin{matrix} (6.30) & C_{1} : = \frac{B_{1} + \sqrt{B_{1}^{2} + α B_{2}}}{α} and C_{2} : = \frac{B_{1} + \sqrt{B_{1}^{2} + α B_{2}}}{α} C_{0} . \end{matrix}$ Using the expressions for $C_{0}$ and α, $B_{1}$ and $B_{2}$ defined at (6.26) and (6.28), respectively, these constants $C_{1}$ and $C_{2}$ can be computed as follows $\begin{array}{l} (6.31) & C_{1} = \frac{(1 + 2 m_{d}) q_{d}}{4 ε γ_{d} (m_{d} - q_{d} - ε)} (1 + \sqrt{1 + 8 q_{d} ε γ_{d}^{2} (m_{d} - q_{d} - ε)}), \\ (6.32) & \begin{matrix} C_{2} = & \frac{| Ω | C_{1}}{q_{d} {(1 + 2 m_{d})}^{2}} {\sum_{i = 1}^{d} [\frac{ν_{i} (q_{i} - p_{i}) (1 + 2 m_{d})}{q_{i} γ_{d}} {(\frac{ν_{i} p_{i}}{q_{d} κ_{i} γ_{d}})}^{\frac{p_{i}}{q_{i} - p_{i}}} \\ + \frac{m_{d} (m_{d} q_{i} - 2 q_{i} - 2 m_{d})}{q_{i}} {(\frac{d^{q_{i}}}{4 q_{d}^{2} κ_{i}^{2}})}^{\frac{m_{d}}{m_{d} q_{i} - 2 q_{i} - 2 m_{d}}} \\ + \frac{4 m_{d} q_{d} κ_{i} (q_{i} - p_{i})}{q_{i} (2 p_{i} - q_{i})} {(\frac{ν_{i}^{2} {(m_{d} - p_{i})}^{2} (2 p_{i} - q_{i})}{2 m_{d} q_{d} κ_{i}^{2}})}^{\frac{q_{i}}{2 (q_{i} - p_{i})}}] \\ + \sum_{i = 1}^{d - 1} \frac{m_{d} - 2 (m_{i} - 1)}{2 (d - 1) (m_{i} - 1)} {(\frac{2 (m_{i} - 1) {((d - 1) γ_{i} (m_{d} - m_{i}))}^{2}}{m_{d}})}^{\frac{m_{d}}{m_{d} - 2 (m_{i} - 1)}}}, \end{matrix} \end{array}$ and where ε is to be chosen according to (6.29).

Hence, by Lemma 2.3, the blow-up property (6.6) holds true for $\begin{matrix} (6.33) & \begin{matrix} T_{max} ⩽ & \frac{1}{2 \sqrt{B_{1}^{2} + α B_{2}}} \\ \times ln (\frac{(- B_{1} + \sqrt{B_{1}^{2} + α B_{2}}) (Φ (0) + C_{0}) + α \sum_{i = 1}^{d} (γ_{i} ‖ u_{0} ‖_{L^{m_{i}} (Ω)}^{m_{i}} - ν_{i} ‖ D_{i} u_{0} ‖_{L^{p_{i}} (Ω)}^{p_{i}})}{(- B_{1} - \sqrt{B_{1}^{2} + α B_{2}}) (Φ (0) + C_{0}) + α \sum_{i = 1}^{d} (γ_{i} ‖ u_{0} ‖_{L^{m_{i}} (Ω)}^{m_{i}} - ν_{i} ‖ D_{i} u_{0} ‖_{L^{_{i}} (Ω)}^{p_{i}})}), \end{matrix} \end{matrix}$ where again $C_{0}$ and α, $B_{1}$ and $B_{2}$ are the positive constants defined at (6.26) and (6.28), respectively. □

7. Blow-up in the case $q_{i} = 2$

In this final section, we consider the problem (1.3)–(1.6) in the case of $f = 0$ and considering a linear and isotropic relaxation term, i.e. we assume that in the equation (1.3) $\begin{matrix} (7.1) & q_{i} = 2, κ_{i} = κ > 0 \forall i \in {1, \dots, d} . \end{matrix}$ Existence and uniqueness results for the problem (1.3)–(1.6), under the assumption (7.1), have been proved in Theorems 7.1 and 8.1 of ours previous work [4]. In view of the assumption (7.1), the notations (3.5), (6.7), and (6.8) can now be written as $\begin{array}{l} Φ (t) : = \frac{1}{2} ({‖ u (t) ‖}_{L^{2} (Ω)}^{2} + κ {‖ \nabla u (t) ‖}_{L^{2} (Ω)}^{2}), \\ I (t) : = \frac{1}{2} {‖ u (t) ‖}_{L^{2} (Ω)}^{2} + \frac{κ}{2} {‖ \nabla u (t) ‖}_{L^{2} (Ω)}^{2} + C_{0}, \\ J (t) : = {‖ u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + κ {‖ \nabla u_{t} (t) ‖}_{L^{2} (Ω)}^{2}, \end{array}$ where $C_{0}$ is the constant defined in (6.26) with the simplifications derived from assuming (7.1). In particular, taking $t = 0$ and observing (1.5), we have $\begin{matrix} Φ (0) = \frac{1}{2} (‖ u_{0} ‖_{L^{2} (Ω)}^{2} + κ ‖ \nabla u_{0} ‖_{L^{2} (Ω)}^{2}) . \end{matrix}$

Theorem 7.1.

Let $ν_{i}$ , $γ_{i} > 0$ for all $i \in {1, \dots, d}$ , and assume that ( 7.1 ) holds together with $\begin{matrix} 1 < p_{i} < 2, m_{d} ⩾ 2 (m_{i} - 1) \forall i \in {1, \dots, d - 1} . \end{matrix}$ In addition, assume that $\begin{matrix} \sum_{i = 1}^{d} γ_{i} ‖ u_{0} ‖_{L^{m_{i}} (Ω)}^{m_{i}} ⩾ \sum_{i = 1}^{d} ν_{i} ‖ D_{i} u_{0} ‖_{L^{p_{i}} (Ω)}^{p_{i}} + C_{1} Φ (0) + C_{2} \end{matrix}$ for the constants $C_{1}$ and $C_{2}$ defined at ( 6.31 )–( 6.32 ) and whose writing can be simplified by virtue of ( 7.1 ).

Then there exists a finite time $T_{max} < \infty$ such that the strong solution to the problem ( 1.3 )–( 1.6 ) blows up in the following sense $\begin{matrix} \frac{1}{2} {‖ u (t) ‖}_{L^{2} (Ω)}^{2} + \frac{κ}{2} {‖ \nabla u (t) ‖}_{L^{2} (Ω)}^{2} ⟶ \infty as t \to T_{max}, \end{matrix}$

Proof.

The proof follows exactly the same approach of the proof of Theorem 6.1, with the simplifications underlying the assumption (7.1), i.e. $q_{i} = 2$ and $κ_{i} = κ$ for all $i \in {1, \dots, d}$ .

The main difference with the proof of Theorem 6.1, relies in the estimate of the term $J_{3} (t)$ established in (6.18). In the present case, we estimate this term as follows $\begin{array}{rcl} | J_{3} (t) | & = & | - \int_{Ω} (u (t) \cdot \nabla) u (t) \cdot u_{t} (t) d x | \\ = & | \int_{Ω} (u (t) \cdot \nabla) u_{t} (t) \cdot u (t) d x | \\ ⩽ & {‖ \nabla u_{t} (t) ‖}_{L^{2} (Ω)} {‖ u (t) ‖}_{L^{4} (Ω)}^{2} \\ ⩽ & \frac{ε}{2 m_{d}} κ {‖ \nabla u_{t} (t) ‖}_{L^{2} (Ω)}^{2} + \frac{1}{2 m_{d} ε} {‖ u (t) ‖}_{L^{m_{d}} (Ω)}^{m_{d}} + J_{33}, \end{array}$ where now $\begin{matrix} J_{33} : = \frac{(m_{d} - 4) | Ω |}{8 m_{d} ε} {(\frac{4 m_{d}}{κ})}^{\frac{m_{d}}{m_{d} - 4}} . \end{matrix}$ Other difference relies on the choice of the constants $ε_{i}$ in the application of Young’s inequality in (6.22). In the present situation, for each $i \in {1, \dots, d}$ , we choose $\begin{matrix} ε_{i} = {(\frac{q_{d} κ_{i}^{2}}{{(ν_{i} | p_{i} - m_{d} |)}^{2} (2 p_{i} - q_{i})})}^{\frac{2 p_{i} - q_{i}}{q_{i}}}, with q_{i} = 2 and κ_{i} = κ \forall i \in {1, \dots, d} . \end{matrix}$ Then, arguing as we did in the proof of Theorem 6.1, we get the inequality (6.27), in this case with $\begin{matrix} α : = \frac{m_{d} - ε}{2} - 1, B_{1} : = \frac{1}{2 ε γ_{d}}, B_{2} : = \frac{2}{ε}, and ε \in (0, m_{d} - 2) . \end{matrix}$

Therefore Lemma 2.3 applies in this case as well. Here, the constants $C_{1}$ and $C_{2}$ are also defined by (6.30), but with the simplifications underlying to the assumption (7.1). Explicit formulae for $C_{1}$ and $C_{2}$ can be obtained by the application of assumption (7.1) to (6.31) and (6.32). In particular, the constant $C_{0}$ that appears in (6.32), can be depicted from (6.24) and (6.26) as follows $\begin{array}{l} C_{0} : = & \frac{| Ω |}{4} {\sum_{i = 1}^{d} [\frac{κ (2 - p_{i})}{p_{i}} {(\frac{ν_{i} p_{i}}{κ γ_{d}})}^{\frac{2}{2 - p_{i}}} + \frac{κ (2 - p_{i})}{p_{i} - 1} {(\frac{ν_{i}^{2} {(m - p_{i})}^{2} (p_{i} - 1)}{κ^{2}})}^{\frac{1}{2 - p_{i}}} \\ + \frac{(m - 4)}{4} {(\frac{4 m_{d}}{κ})}^{\frac{m_{d}}{m_{d} - 4}}] \\ + \sum_{i = 1}^{d - 1} \frac{(m_{d} - 2 (m_{i} - 1))}{2 (d - 1) (m_{i} - 1)} {(\frac{2 (m_{i} - 1) {((d - 1) γ_{i} (m_{d} - m_{i}))}^{2}}{m_{d}})}^{\frac{m_{d}}{m_{d} - 2 (m_{i} - 1)}}} . \end{array}$ This concludes the proof. □

Footnotes

Acknowledgements

The first author was supported by the Research Project 19-11-00069 of the Russian Science Foundation, Russia (30 percent of all results of this paper). The second author was supported by the Grant no. SFRH/BSAB/135242/2017 of the Portuguese Foundation for Science and Technology, Portugal. Both first and second authors were also partially supported by National Funding from FCT – Fundação para a Ciência e a Tecnologia, under the project: UIDB/04561/2020. The third author was partially supported by the Grant no. AP08052425 of the Ministry of Science and Education of the Republic of Kazakhstan (MES RK), Kazakhstan.

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