Existence and asymptotic behavior of a parabolic equation with L 1 data

Abstract

We study a parabolic equation with the data and the coefficient of zero order term only summable functions. Despite all this lack of regularity we prove that there exists a solution which becomes immediately bounded. Moreover, we study the asymptotic behavior of this solution in the autonomous case showing that the constructed solution tends to the associate stationary solution.

Keywords

Regularizing effect asymptotic behavior regularity of solutions linear parabolic equations

1. Introduction and statement of results

In this paper we study the following linear parabolic problem $\begin{matrix} (1.1) & \{\begin{matrix} u_{t} - div (M (x, t) \nabla u) + a (x, t) u = f (x, t) & in Ω_{T} = Ω \times (0, T), \\ u (x, t) = 0 & on Γ_{T} = \partial Ω \times (0, T), \\ u (x, 0) = u_{0} (x) & in Ω, \end{matrix} \end{matrix}$ where Ω is a bounded open subset of $R^{N}$ ( $N > 2$ ), T is a positive constant and M is a bounded and uniformly elliptic matrix, i.e. there exist $α > 0$ and $β > 0$ such that $\begin{matrix} (1.2) & M (x, t) ξ \cdot ξ ⩾ α | ξ |^{2}, | M (x, t) | ⩽ β, \end{matrix}$ for almost every $(x, t)$ in $Ω_{T}$ and for every ξ in $R^{N}$ . Furthermore, on the functions $a (x, t)$ and $f (x, t)$ we assume $\begin{matrix} (1.3) & a (x, t) ⩾ 0, a (x, t) \in L^{1} (Ω_{T}), f (x, t) \in L^{1} (Ω_{T}), \end{matrix}$ and that there exists a non negative constant Q such that $\begin{matrix} (1.4) & | f (x, t) | ⩽ Q a (x, t) . \end{matrix}$ Finally, on the initial datum $u_{0}$ we suppose that $\begin{matrix} (1.5) & u_{0} \in L^{1} (Ω) . \end{matrix}$ Here, by a weak solution we mean a function u belonging to $L^{1} (0, T; W_{0}^{1, 1} (Ω))$ satisfying $a (x, t) u \in L^{1} (Ω_{T})$ and such that $\begin{matrix} (1.6) & \begin{array}{l} \iint_{Ω_{T}} {- u φ_{t} + M (x, t) \nabla u \cdot \nabla φ + a (x, t) u φ} \\ = \iint_{Ω_{T}} f (x, t) φ + \int_{Ω} u_{0} φ (0), \end{array} \end{matrix}$ for every φ belonging to $W^{1, \infty} (0, T; L^{\infty} (Ω)) \cap L^{\infty} (0, T; W_{0}^{1, \infty} (Ω))$ satisfying $φ (T) = 0$ .

We prove that the previous structure assumptions guarantee the existence of at least a weak solution u of problem (1.1) that becomes “immediately bounded”, i.e. such that $\begin{matrix} u \in L^{\infty} (Ω \times (ε, T)) for every ε > 0 . \end{matrix}$ Moreover, if $u_{0} \in L^{\infty} (Ω)$ then u is bounded up to $t = 0$ (see Remark 3.1 below for further regularity properties and details).

Hence the linear equation (1.1) has a very strong regularizing effect since, although all the data $u_{0}$ , $a (x, t)$ and $f (x, t)$ are assumed to be only summable functions, there exists a solution u that becomes immediately bounded.

This kind of regularizing phenomena are well known to appear in absence of the irregular data f and a (see for example [10,23,24] and [16]). On the other side, it is also true that this regularization does not appear if we replace the previous assumptions on the data a and f with $\begin{matrix} a (x, t) \equiv 0, f \in L^{1} (Ω_{T}) . \end{matrix}$ Hence, the presence of the lower order term $a (x, t) u$ combined with the control condition in (1.4) causes this “instantaneous boundedness”.

Besides, it is also possible to estimate the $L^{\infty}$ -norm of this solution by a constant depending only on the data as the following estimate shows $\begin{matrix} (1.7) & {‖ u (t) ‖}_{L^{\infty} (Ω)} ⩽ C_{1} max {\frac{‖ u_{0} ‖_{L^{1} (Ω)}}{t^{\frac{N}{2}}}, Q} a.e. t \in (0, T), \end{matrix}$ where $C_{1}$ depends only on the coerciveness coefficient α in (1.2) and on N (see Theorem 3.1 below). It is interesting to notice that for $Q = 0$ the previous estimate becomes the well known decay estimate $\begin{matrix} (1.8) & {‖ u (t) ‖}_{L^{\infty} (Ω)} ⩽ C_{1} \frac{‖ u_{0} ‖_{L^{1} (Ω)}}{t^{\frac{N}{2}}} a.e. t \in (0, T), \end{matrix}$ satisfied not only by the solution of the heat equation $\begin{matrix} \{\begin{matrix} u_{t} - Δ u = 0 & in Ω_{T}, \\ u (x, t) = 0 & on Γ_{T}, \\ u (x, 0) = u_{0} (x) & in Ω, \end{matrix} \end{matrix}$ but also by the solutions of all the slight modification of the heat equation of the following type $\begin{matrix} (1.9) & \{\begin{matrix} u_{t} - div (M (x, t) \nabla u) = 0 & in Ω_{T}, \\ u (x, t) = 0 & on Γ_{T}, \\ u (x, 0) = u_{0} (x) & in Ω, \end{matrix} \end{matrix}$ with M satisfying (1.2) (see [24] if $M (x, t) = M (x)$ , [10] if $u_{0} \in L^{2} (Ω)$ and [16] in the remaining cases), where the constant $C_{1}$ also in this case depends only on the coerciveness coefficient α in (1.2) (and hence it is the same constant for all the problems having the same coerciveness coefficient) (see [16]).

Notice that problem (1.9) corresponds to the particular case $Q = a (x, t) \equiv 0$ in (1.1). Hence there is a sort of “continuity” in these $L^{\infty}$ -estimates passing from the evolution problem (1.1) to the classical parabolic equation (1.9).

Indeed, we prove here that estimate (1.7) holds true for every nonnegative Q. Hence, it follows that estimate (1.8) is satisfied also by the solutions of the following problems $\begin{matrix} (1.10) & \{\begin{matrix} u_{t} - div (M (x, t) \nabla u) + a (x, t) u = 0 & in Ω_{T}, \\ u (x, t) = 0 & on Γ_{T}, \\ u (x, 0) = u_{0} (x) & in Ω, \end{matrix} \end{matrix}$ that is our problem when $Q = 0$ and hence also in this case there is a sort of “continuity” in these $L^{\infty}$ -estimates passing from the evolution problem (1.10) with $a (x, t)$ a nonnegative summable function to the classical case (1.9).

The importance of estimate (1.7) relies in various interesting informations that it gives on the solution. As a matter of fact, not only it shows that the $L^{\infty}$ -norm of the solution $u (t)$ can be controlled by a constant depending only on the $L^{1}$ -norm of the initial datum $u_{0}$ , the dimension N, the coerciveness constant α, the constant Q in (1.4) and the time t, but it also describes the blow-up of the $L^{\infty}$ -norm of $u (t)$ that occurs letting $t \to 0$ if $u_{0} \notin L^{\infty} (Ω)$ , revealing that it cannot exceed the power $t^{- \frac{N}{2}}$ .

In this paper we also investigate the autonomous case $\begin{matrix} (1.11) & \{\begin{matrix} u_{t} - div (M (x) \nabla u) + a (x) u = f (x) & in Ω_{\infty} = Ω \times (0, + \infty), \\ u (x, t) = 0 & on Γ_{\infty} = \partial Ω \times (0, + \infty), \\ u (x, 0) = u_{0} (x) & in Ω, \end{matrix} \end{matrix}$ proving that there exists a global solution u which converges (letting $t \to + \infty$ ) to the unique solution w of the associated stationary problem $\begin{matrix} (1.12) & \{\begin{matrix} - div (M (x) \nabla w) + a (x) w = f (x) & in Ω, \\ w (x) = 0 & on \partial Ω . \end{matrix} \end{matrix}$ We recall that the stationary case (1.12) was studied in [1] where it was proved that if $a (x)$ is a nonnegative summable function and if f is a summable function satisfying the control condition (1.4) then there exists an unique solution u of (1.12) belonging to $H_{0}^{1} (Ω) \cap L^{\infty} (Ω)$ . We observe that elliptic problems with lower order term of the type $a (x) u$ with the coefficients $a (x)$ only summable functions are investigated in [9]. See also [2] and [5]. Moreover, existence results for this kind of stationary problems in Marcinkiewicz space can be found in [4] (see also [3,6,7,12]).

We point out that the structure conditions (1.2) and (1.3) are sufficient conditions to the existence of a weak solution (see Remark 3.3 below) and that in absence of the further assumption (1.4) generally the improvement of regularity described above does not appear.

The paper is organized as follows. In the next section we recall some preliminary results proved in [16] that will be essential in the proofs of the results presented above. In Section 3 we prove the existence of a solution that becomes “immediately bounded” while we study the autonomous case in Section 4.

2. Preliminary results

To be self-contained we recall here two results that, as said above, will be an essential tool in the proof of the stated results.

Theorem 2.1.
Let us assume that $\begin{matrix} (2.1) & v \in C ((0, T); L^{2} (Ω)) \cap L^{2} (0, T; L^{2^{}} (Ω)) \cap C ([0, T); L^{1} (Ω)) \end{matrix}$ where* Ω is an open bounded set of $R^{N}$ with $N ⩾ 3$ and $0 < T ⩽ + \infty$ . Suppose that $σ_{0}$ is a positive constant and that v satisfies for every $k ⩾ σ_{0}$ and $0 < t_{1} < t_{2} < T$ the following inequalities $\begin{array}{l} (2.2) & \int_{Ω} {| G_{k} (v) |}^{2} (t_{2}) d x - \int_{Ω} {| G_{k} (v) |}^{2} (t_{1}) d x + c_{1} \int_{t_{1}}^{t_{2}} {‖ G_{k} (v) (τ) ‖}_{L^{2^{}} (Ω)}^{2} d τ ⩽ 0 \\ (2.3) & {‖ G_{k} (v) (t) ‖}_{L^{1} (Ω)} ⩽ c_{2} {‖ G_{k} (v) (t_{0}) ‖}_{L^{1} (Ω)} for every 0 ⩽ t_{0} < t < T, \end{array}$ where* $c_{1}$ and $c_{2}$ are positive constants independent of k.

Finally, let us define $\begin{matrix} (2.4) & v_{0} \equiv v (x, 0) \in L^{1} (Ω) . \end{matrix}$ Then the following decay estimate occurs $\begin{matrix} {‖ v (t) ‖}_{L^{\infty} (Ω)} ⩽ C_{1} max {σ_{0}, \frac{‖ v_{0} ‖_{L^{1} (Ω)}}{t^{\frac{N}{2}}}} for every t \in (0, T), \end{matrix}$ where $C_{1}$ is a constant depending only on N, $c_{1}$ and $c_{2}$ .

Here the function $G_{k} (s)$ ( $k ⩾ 0$ ) is defined by $\begin{matrix} (2.5) & G_{k} (s) = {(| s | - k)}_{+} sign (s) for every s \in R . \end{matrix}$ The previous result is an immediate consequence of Theorem 2.1 (considered here in the particular case $r = 2$ , $r_{0} = 1$ , $b = 2$ and $q = 2^{}$ ) and Remark 4.1 in [16]. A more general version (including the previous theorem) is proved in Section 3 of [21].
Theorem 2.2.
Let v be as in (* 2.1 ). Assume that ( 2.2 ) and ( 2.3 ) are satisfied for every $k > 0$ . Then the following exponential decay estimate occurs $\begin{matrix} {‖ v (t) ‖}_{L^{\infty} (Ω)} ⩽ C_{2} \frac{‖ v_{0} ‖_{L^{1} (Ω)}}{t^{\frac{N}{2}} e^{σ t}} for every t \in (0, T), \end{matrix}$ where $v_{0}$ is as in ( 2.4 ), $C_{2}$ (see formula (4.17) in [ 16 ]) is a positive constant depending only on N, $c_{1}$ and $c_{2}$ while σ is given by the following formula $\begin{matrix} σ = \frac{c_{1} κ}{4 | Ω |^{\frac{2}{N}}}, κ arbitrarily fixed in (0, \frac{1}{2}), \end{matrix}$ where $| Ω |$ denotes the measure of Ω.

The previous result is a particular case of Theorem 2.2 in [16] (applied here with $r = 2$ , $r_{0} = 1$ , $b = 2$ and $q = 2^{*}$ ).
3. Existence of $L^{\infty} (Ω \times (ε, T))$ solutions ( $ε > 0$ )

In this section we prove the existence of a solution u that becomes “immediately bounded”. Our result is the following.

Theorem 3.1.
Suppose that ( 1.2 )–( 1.5 ) hold true. Then problem ( 1.1 ) has at least a weak solution u satisfying the following regularity properties $\begin{array}{l} (3.1) & u \in L^{q} (0, T; W_{0}^{1, q} (Ω)) \cap L^{\infty} (0, T; L^{1} (Ω)) for every q < 2 - \frac{1}{N + 1} \\ (3.2) & u \in V_{loc} \equiv L_{loc}^{\infty} ((0, T]; L^{\infty} (Ω)) \cap L_{loc}^{2} ((0, T]; H_{0}^{1} (Ω)) . \end{array}$ Finally, estimate ( 1.7 ) holds true.
Proof of Theorem 3.1.
Let $u_{n} \in L^{\infty} (Ω_{T}) \cap L^{2} (0, T; H_{0}^{1} (Ω)) \cap C (0, T; L^{2} (Ω))$ (n positive integers) be the solutions of the following approximating problems $\begin{matrix} (3.3) & \{\begin{matrix} {(u_{n})}_{t} - div (M (x, t) \nabla u_{n}) + a_{n} (x, t) u_{n} = f_{n} & in Ω_{T}, \\ u_{n} (x, t) = 0 & on Γ_{T}, \\ u_{n} (x, 0) = u_{0, n} (x) & in Ω . \end{matrix} \end{matrix}$ where $a_{n}$ and $f_{n}$ are bounded functions defined as follows $\begin{matrix} (3.4) & f_{n} (x, t) = \frac{f (x, t)}{1 + \frac{1}{n} | f (x, t) |}, a_{n} (x, t) = \frac{a (x, t)}{1 + \frac{Q}{n} | a (x, t) |}, \end{matrix}$ and $u_{0, n} (x) \in C^{1} (\bar{Ω})$ satisfies $\begin{matrix} ‖ u_{0, n} ‖_{L^{1} (Ω)} ⩽ ‖ u_{0} ‖_{L^{1} (Ω)} u_{0, n} \to u_{0} in L^{1} (Ω), \end{matrix}$ (see Theorems 4.1, 4.2 and 7.1 in chapter III of [13]).

Observe that, since $ψ (s) = s {(1 + \frac{s}{n})}^{- 1}$ is an increasing function, by assumption (1.4) we deduce $\begin{matrix} (3.5) & | f_{n} (x, t) | ⩽ \frac{| f (x, t) |}{1 + \frac{1}{n} | f (x, t) |} ⩽ \frac{Q a (x, t)}{1 + \frac{Q}{n} a (x, t)} = Q a_{n} (x, t) \end{matrix}$ The proof proceeds by steps.

Step 1. We prove in this step that the sequence ${u_{n}}$ satisfies the following inequality $\begin{matrix} (3.6) & | u_{n} (x, t) | ⩽ A (t) \equiv C_{1} max {\frac{‖ u_{0} ‖_{L^{1} (Ω)}}{t^{\frac{N}{2}}}, Q} for every (x, t) \in Ω_{T}, \end{matrix}$ where $C_{1}$ depends only on α (in (1.2)) and N.

In order to prove the previous estimate, let $t_{0}$ and $t_{1}$ be arbitrarily fixed positive numbers satisfying $0 < t_{0} < t_{1} ⩽ T$ . Let $k ⩾ Q$ and choose $G_{k} (u_{n})$ as a test function in (3.3) (see (2.5) for the definition of $G_{k} (\cdot)$ ). We obtain, using (1.2) and (3.5) $\begin{array}{l} \frac{1}{2} \int_{Ω} {| G_{k} (u_{n}) (t_{1}) |}^{2} - \frac{1}{2} \int_{Ω} {| G_{k} (u_{n}) (t_{0}) |}^{2} + α \int_{t_{0}}^{t_{1}} \int_{Ω} {| \nabla G_{k} (u_{n}) |}^{2} \\ + \int_{t_{0}}^{t_{1}} \int_{Ω} a_{n} (x, t) u_{n} G_{k} (u_{n}) ⩽ \int_{t_{0}}^{t_{1}} \int_{Ω} Q a_{n} (x, t) | G_{k} (u_{n}) |, \end{array}$ which, thanks to the choice $k ⩾ Q$ , implies $\begin{matrix} \frac{1}{2} \int_{Ω} {| G_{k} (u_{n}) (t_{1}) |}^{2} - \frac{1}{2} \int_{Ω} {| G_{k} (u_{n}) (t_{0}) |}^{2} + α \int_{t_{0}}^{t_{1}} \int_{Ω} {| \nabla G_{k} (u_{n}) |}^{2} ⩽ 0 . \end{matrix}$ Thus, by Sobolev inequality we get for every $0 < t_{0} < t_{1} ⩽ T$ and $k ⩾ Q$ $\begin{matrix} (3.7) & \frac{1}{2} \int_{Ω} {| G_{k} (u_{n}) (t_{1}) |}^{2} - \frac{1}{2} \int_{Ω} {| G_{k} (u_{n}) (t_{0}) |}^{2} + α c_{S} \int_{t_{0}}^{t_{1}} {‖ G_{k} (u_{n}) ‖}_{2^{*}}^{2} ⩽ 0 . \end{matrix}$ Let $δ > 1$ and as above $k ⩾ Q$ arbitrarily fixed. Taking $φ = {1 - \frac{1}{{(1 + | G_{k} (u_{n}) |)}^{δ}}} sign (u_{n})$ as a test function in (3.3) and using assumption (1.2) we obtain $\begin{array}{l} \int_{Ω} | G_{k} (u_{n}) | (t) + δ α \int_{t_{0}}^{t} \int_{Ω} \frac{| \nabla G_{k} (u_{n}) |^{2}}{{(1 + | G_{k} (u_{n}) |)}^{δ + 1}} \\ - \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | G_{k} (u_{n}) | (t))}^{δ - 1}}} + \int_{t_{0}}^{t} \int_{Ω} a_{n} (x, t) | u_{n} | {1 - \frac{1}{{(1 + | G_{k} (u) |)}^{δ}}} \\ ⩽ \int_{Ω} | G_{k} (u_{n}) | (t_{0}) - \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | G_{k} (u_{n}) | (t_{0}))}^{δ - 1}}} \\ (3.8) & + \int_{t_{0}}^{t} \int_{Ω} | f_{n} (x, t) | {1 - \frac{1}{{(1 + | G_{k} (u_{n}) |)}^{δ}}} . \end{array}$ Hence using again estimate (3.5) by (3.8) we deduce $\begin{array}{l} \int_{Ω} | G_{k} (u_{n}) | (t) + δ α \int_{t_{0}}^{t} \int_{Ω} \frac{| \nabla G_{k} (u_{n}) |^{2}}{{(1 + | G_{k} (u_{n}) |)}^{δ + 1}} \\ - \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | G_{k} (u_{n}) | (t))}^{δ - 1}}} \\ + \int_{t_{0}}^{t} \int_{Ω} a_{n} (x, t) (| u_{n} | - Q) {1 - \frac{1}{{(1 + | G_{k} (u) |)}^{δ}}} \\ (3.9) & ⩽ \int_{Ω} | G_{k} (u_{n}) | (t_{0}) - \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | G_{k} (u_{n}) | (t_{0}))}^{δ - 1}}} . \end{array}$ Observe that since $k ⩾ Q$ it results $\begin{matrix} \int_{t_{0}}^{t} \int_{Ω} a_{n} (x, t) (| u_{n} | - Q) {1 - \frac{1}{{(1 + | G_{k} (u_{n}) |)}^{δ}}} ⩾ 0 . \end{matrix}$ Thus, dropping the positive terms in (3.9) we deduce $\begin{matrix} \int_{Ω} | G_{k} (u_{n}) | (t) ⩽ \int_{Ω} | G_{k} (u_{n}) | (t_{0}) + \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | G_{k} (u_{n}) | (t))}^{δ - 1}}} \end{matrix}$ which implies $\begin{matrix} \int_{Ω} | G_{k} (u_{n}) | (t) ⩽ \int_{Ω} | G_{k} (u_{n}) | (t_{0}) + \frac{| Ω |}{δ - 1} . \end{matrix}$ Letting $δ \to + \infty$ , we obtain that for every $k ⩾ Q$ and $0 ⩽ t_{0} < t ⩽ T$ $\begin{matrix} (3.10) & {‖ G_{k} (u_{n}) (t) ‖}_{L^{1} (Ω)} ⩽ {‖ G_{k} (u_{n}) (t_{0}) ‖}_{L^{1} (Ω)} . \end{matrix}$ By inequalities (3.7) and (3.10) it follows that we can apply Theorem 2.1 (with $σ_{0} = Q$ ) which allows to conclude that estimate (3.6) holds true.

Step 2. Let $t_{0}$ be a number arbitrarily fixed such that $0 < t_{0} < T$ and define $Ω_{T}^{t_{0}} = Ω \times (t_{0}, T)$ . We show in this step that the following estimate holds $\begin{matrix} (3.11) & ‖ u_{n} ‖_{L^{\infty} (t_{0}, T; L^{2} (Ω))}^{2} + ‖ \nabla u_{n} ‖_{L^{2} (Ω_{T}^{t_{0}})}^{2} ⩽ min {\frac{1}{2}, α}^{- 1} A (t_{0}) [‖ f ‖_{L^{1} (Ω_{T})} + \frac{| Ω |}{2} A (t_{0})], \end{matrix}$ where $A (\cdot)$ is as in (3.6). To prove (3.11) take $u_{n}$ as a test function in (3.3). Then, using (1.2) and (3.6) (and dropping positive terms) we have for every $0 < t_{0} < t ⩽ T$ $\begin{matrix} \frac{1}{2} \int_{Ω} u_{n}^{2} (t) - \frac{1}{2} \int_{Ω} u_{n}^{2} (t_{0}) + α \int_{t_{0}}^{t} \int_{Ω} | \nabla u_{n} |^{2} ⩽ A (t_{0}) ‖ f ‖_{L^{1} (Ω_{T})} . \end{matrix}$ Now (3.11) follows by the previous estimate using again (3.6).

A consequence of (3.11) is that the sequence ${u_{n}}$ is bounded in the space $L^{2} (t_{0}, T; H_{0}^{1} (Ω))$ . In particular, up to subsequences (still denoted $u_{n}$ ) we have that $u_{n}$ weakly converges to some function u in $L^{2} (t_{0}, T; H_{0}^{1} (Ω))$ .

Step 3. Let $t_{0}$ be a positive number arbitrarily fixed such that $t_{0} ⩽ T$ . Then, recalling that $u_{n}$ solves (3.3) and using estimates (3.6) and (3.11) proved in the previous steps we deduce that the sequence ${{(u_{n})}_{t}}$ is bounded in the space $L^{2} (t_{0}, T; H^{- 1} (Ω)) + L^{1} (Ω_{T}^{t_{0}})$ . By the classical result due to Simon (see [22]), this implies that (up to subsequences) $u_{n}$ strongly converges to u in $L^{1} (Ω_{T}^{t_{0}})$ and almost everywhere in $Ω_{T}^{t_{0}}$ . Being $u_{n}$ equibounded in $Ω_{T}^{t_{0}}$ it follows that $u_{n}$ converges also in $L^{2} (Ω_{T}^{t_{0}})$ .

Furthermore, since $t_{0}$ is an arbitrary positive number, it follows that $u_{n}$ converges to u almost everywhere in $Ω_{T}$ and again by (3.6) it follows that $u \in L_{loc}^{\infty} (0, T; L^{\infty} (Ω))$ and satisfies (1.7).

Using the above a priori estimates and convergence results, we can pass to the limit in (3.3) obtaining that the function u satisfies $\begin{matrix} \iint_{Ω_{T}} {- u φ_{t} + M (x, t) \nabla u \cdot \nabla φ + a (x, t) u φ} = \iint_{Ω_{T}} f (x, t) φ for every φ \in C_{c}^{\infty} (Ω_{T}) \end{matrix}$ Hence u is a distributional solution of (1.1). It remains to show that u satisfies “in some weak sense” also the initial condition “ $u (x, 0) = u_{0}$ ”‘. We prove it showing that the function u satisfies also $\begin{array}{l} \iint_{Ω_{T}} {- u φ_{t} + M (x, t) \nabla u \cdot \nabla φ + a (x, t) u φ} \\ = \iint_{Ω_{T}} f (x, t) φ + \int_{Ω} u_{0} φ (0) for every φ \in C^{\infty} ({\bar{Ω}}_{T}) : φ (T) = 0, \end{array}$ i.e. u is a weak solution of (1.1) according to the definition (1.6) given above. In order to do it we need to improve the above a priori estimates.

Step 4. We show here that the sequence ${u_{n}}$ satisfies the following two estimates $\begin{matrix} (3.12) & ‖ u_{n} ‖_{L^{\infty} (0, T; L^{1} (Ω))} ⩽ ‖ u_{0} ‖_{L^{1} (Ω)} + ‖ f ‖_{L^{1} (Ω_{T})}, \end{matrix}$ and $\begin{matrix} (3.13) & δ α \int_{0}^{T} \int_{Ω} \frac{| \nabla u_{n} |^{2}}{{(1 + | u_{n} |)}^{δ + 1}} ⩽ C_{0} for every δ \in (0, 1), \end{matrix}$ where $C_{0} = (1 + \frac{1}{1 - δ}) [‖ u_{0} ‖_{L^{1} (Ω)} + ‖ f ‖_{L^{1} (Ω_{T})}]$ .

To prove these estimates let $δ > 0$ and take $\begin{matrix} φ = (1 - \frac{1}{{(1 + | u_{n} |)}^{δ}}) sign (u_{n}) \end{matrix}$ as a test function in (3.3). Using assumption (1.2) (and the fact that $| φ | ⩽ 1$ ) we obtain $\begin{matrix} (3.14) & \begin{matrix} \int_{Ω} | u_{n} | (t) + δ α \int_{0}^{t} \int_{Ω} \frac{| \nabla u_{n} |^{2}}{{(1 + | u_{n} |)}^{δ + 1}} \\ - \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | u_{n} | (t))}^{δ - 1}}} + \int_{0}^{t} \int_{Ω} a_{n} (x, t) | u_{n} | (1 - \frac{1}{{(1 + | u_{n} |)}^{δ}}) \\ ⩽ \int_{Ω} | u_{0} | - \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | u_{0} |)}^{δ - 1}}} + ‖ f ‖_{L^{1} (Ω_{T})} . \end{matrix} \end{matrix}$ Hence, dropping the positive term coming from the lower order term by (3.14) we deduce $\begin{matrix} (3.15) & \begin{matrix} \int_{Ω} & | u_{n} | (t) + δ α \int_{0}^{t} \int_{Ω} \frac{| \nabla u_{n} |^{2}}{{(1 + | u_{n} |)}^{δ + 1}} - \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | u_{n} | (t))}^{δ - 1}}} \\ ⩽ \int_{Ω} | u_{0} | - \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | u_{0} |)}^{δ - 1}}} + ‖ f ‖_{L^{1} (Ω_{T})} . \end{matrix} \end{matrix}$

First of all, choosing $δ > 1$ , from (3.15) we deduce $\begin{matrix} (3.16) & \int_{Ω} | u_{n} | (t) ⩽ \int_{Ω} | u_{0} | + \frac{| Ω |}{δ - 1} + ‖ f ‖_{L^{1} (Ω_{T})}, \end{matrix}$ which, letting $δ \to + \infty$ , implies (3.12).

Now, by (3.12), we deduce that for every $δ \in (0, 1)$ $\begin{matrix} \frac{1}{δ - 1} \int_{Ω} {1 - \frac{1}{{(1 + | u_{n} | (t))}^{δ - 1}}} & = \frac{1}{1 - δ} \int_{Ω} {{(1 + | u_{n} | (t))}^{1 - δ} - 1} \\ ⩽ \frac{1}{1 - δ} \int_{Ω} | u_{n} | (t) ⩽ \frac{1}{1 - δ} (\int_{Ω} | u_{0} | + ‖ f ‖_{L^{1} (Ω_{T})}) . \end{matrix}$ By the previous estimate and (3.15) it follows that for every $δ \in (0, 1)$ $\begin{matrix} δ α \int_{0}^{t} \int_{Ω} \frac{| \nabla u_{n} |^{2}}{{(1 + | u_{n} |)}^{δ + 1}} \\ ⩽ \int_{Ω} | u_{0} | + ‖ f ‖_{L^{1} (Ω_{T})} + \frac{1}{1 - δ} (\int_{Ω} | u_{0} | + ‖ f ‖_{L^{1} (Ω_{T})}), \end{matrix}$ which implies (3.13).

Step 5. We prove in this step further convergence results. By estimates (3.12) and (3.13) proved in Step 4 and Corollary 3.1 in [17] we deduce that there exists a positive constant $C_{3}$ (independent of n) such that $\begin{matrix} \int_{0}^{T} \int_{Ω} | u_{n} |^{1 - δ + \frac{2}{N}} ⩽ C_{3} for every δ \in (0, 1) . \end{matrix}$ Choosing δ such that $0 < δ < \frac{2}{N}$ , we have that $m = 1 - δ + \frac{2}{N} > 1$ and, up to a subsequence, $\begin{matrix} u_{n} weakly converges to u in L^{m} (Ω_{T}) . \end{matrix}$ Therefore, since $u_{n}$ converges to u almost everywhere in $Ω_{T}$ , we obtain $\begin{matrix} u_{n} strongly converges to u in L^{1} (Ω_{T}) . \end{matrix}$ Recall that the convergence proved in Step 3 was far from $t = 0$ . Furthermore, by the estimates proved in Step 4 and Lemma 3.2 in [17] it follows that there exists a positive constant $C_{4}$ (independent of n) such that $\begin{matrix} ‖ \nabla u_{n} ‖_{L^{s} (Ω_{T})} ⩽ C_{4}, \end{matrix}$ where $\begin{matrix} s = max {\frac{2}{2 + δ}, 2 - \frac{N (1 + δ)}{N + 1}} for every δ \in (0, 1) . \end{matrix}$ Note that, for $δ < \frac{N}{2}$ we have $\begin{matrix} max {\frac{2}{2 + δ}, 2 - \frac{N (1 + δ)}{N + 1}} = 2 - \frac{N (1 + δ)}{N + 1} . \end{matrix}$ As a consequence, we obtain that (up to a subsequence) $\begin{matrix} \nabla u_{n} weakly converges to \nabla u in L^{q} (Ω_{T}) for every q < 2 - \frac{N}{N + 1} . \end{matrix}$ We notice that now we can pass to the limit (as $n \to + \infty$ ) in all the terms in the weak formulation satisfied by $u_{n}$ except in $\int_{Ω_{T}} a_{n} (x, t) u_{n} φ$ . Hence, in order to prove that it is possible to pass to the limit also in this last term we only have to prove that $\begin{matrix} (3.17) & lim_{n \to \infty} \iint_{Ω_{T}} a_{n} (x, t) u_{n} φ = \iint_{Ω_{T}} a (x, t) u φ for every φ \in C^{\infty} (\overline{Ω_{T}}) \end{matrix}$ and the idea is to apply Vitali’s theorem. The proof of (3.17) is in Step 8 below and make use of the results in the following Steps 6 and Step 7.

Step 6. We prove in this step that the sequence ${a_{n} (x, t) u_{n} (x, t)}$ is bounded in $L^{1} (Ω_{T})$ and the following estimate holds true $\begin{matrix} \iint_{Ω_{T}} a (x, t) | u (x, t) | ⩽ ‖ u_{0} ‖_{L^{1} (Ω)} + ‖ f ‖_{L^{1} (Ω_{T})} . \end{matrix}$ As a matter of factk, let $k > 0$ arbitrarily fixed and take $\frac{T_{k} (u_{n})}{k}$ as a test function in (3.3), where $T_{k} (s)$ is the usual truncation at level k defined as $\begin{matrix} T_{k} (s) = min {k, max {s, - k}} . \end{matrix}$ Using assumption (1.2) and dropping the positive terms we deduce $\begin{matrix} \iint_{Ω_{T}} a_{n} (x, t) u_{n} (x, t) \frac{T_{k} (u_{n} (x, t))}{k} ⩽ ‖ f ‖_{L^{1} (Ω_{T})} + ‖ u_{0} ‖_{L^{1} (Ω)}, \end{matrix}$ which implies, using Fatou’s Lemma letting $k \to + \infty$ $\begin{matrix} \iint_{Ω_{T}} a_{n} (x, t) | u_{n} (x, t) | ⩽ ‖ f ‖_{L^{1} (Ω_{T})} + ‖ u_{0} ‖_{L^{1} (Ω)} . \end{matrix}$ Since, the sequence ${u_{n}}$ converges almost everywhere in $Ω_{T}$ , Fatou’s Lemma implies again $\begin{matrix} \iint_{Ω_{T}} a (x, t) | u (x, t) | ⩽ ‖ f ‖_{L^{1} (Ω_{T})} + ‖ u_{0} ‖_{L^{1} (Ω)}, \end{matrix}$ that is the assertion.

Step 7. We prove now that the sequence ${a_{n} (x, t) u_{n} (x, t)}$ is equi-integrable.

To this aim, let us define $\begin{matrix} G_{k, k + ϵ} (s) = \{\begin{matrix} - 1 & s ⩽ - k - ϵ \\ \frac{s + k}{ϵ} & - k - ϵ ⩽ s ⩽ - k \\ 0 & s \in (- k, k) \\ \frac{s - k}{ϵ} & k ⩽ s ⩽ k + ϵ \\ 1 & s ⩾ k + ϵ \end{matrix} \end{matrix}$ and take $φ = G_{k, k + ϵ} (u_{n} (x, t))$ as a test function in (3.3). We get $\begin{array}{l} \int_{Ω} Φ (u_{n} (x, T)) + α \iint_{Ω_{T}} {| \nabla G_{k, k + ϵ} (u_{n} (x, t)) |}^{2} + \iint_{Ω_{T}} a_{n} (x, t) u_{n} (x, t) G_{k, k + ϵ} (u_{n} (x, t)) \\ (3.18) & = \iint_{Ω_{T}} f_{n} (x, t) G_{k, k + ϵ} (u_{n} (x, t)) + \int_{Ω} Φ (u_{0, n} (x)), \end{array}$ where $\begin{matrix} Φ (s) = \int_{0}^{s} G_{k, k + ϵ} (σ) d σ . \end{matrix}$ Observe that it results $\begin{matrix} \int_{Ω} Φ (u_{n} (x, T) ⩾ 0 \end{matrix}$ and $\begin{matrix} \int_{Ω} Φ (u_{0, n} (x)) ⩽ ‖ u_{0, n} ‖_{L^{1} (A_{k, 0}^{n})}, A_{k, 0}^{n} = {x \in Ω : | u_{0, n} (x) | > k} . \end{matrix}$ Thus, from the (3.18) (dropping the positive terms) we deduce $\begin{matrix} \iint_{Ω_{T}} a_{n} (x, t) u_{n} (x, t) G_{k, k + ϵ} (u_{n} (x, t)) ⩽ ‖ f ‖_{L^{1} (A_{k}^{n})} + ‖ u_{0, n} ‖_{L^{1} (A_{k, 0}^{n})}, \end{matrix}$ where $\begin{matrix} A_{k}^{n} = {(x, t) \in Ω_{T} : | u_{n} (x, t) | > k} . \end{matrix}$ Therefore letting $ϵ \to 0$ we obtain $\begin{matrix} (3.19) & \iint_{A_{k}^{n}} a_{n} (x, t) | u_{n} (x, t) | ⩽ ‖ f ‖_{L^{1} (A_{k}^{n})} + ‖ u_{0, n} ‖_{L^{1} (A_{k, 0}^{n})} . \end{matrix}$ Then, for every measurable subset E of $Ω_{T}$ , thanks to the above inequality we have $\begin{array}{l} \iint_{E} a_{n} (x, t) | u_{n} (x, t) | & ⩽ \iint_{A_{k}^{n}} a_{n} (x, t) | u_{n} (x, t) | + k \iint_{E} a (x, t) \\ ⩽ ‖ f ‖_{L^{1} (A_{k}^{n})} + ‖ u_{0, n} ‖_{L^{1} (A_{k, 0}^{n})} + k \iint_{E} a (x, t) \end{array}$ which implies the result taking into account that $u_{0, n}$ strongly converges to $u_{0}$ in $L^{1} (Ω)$ .

Step 8. Since the sequence ${a_{n} (x, t) u_{n} (x, t)}$ is equi-integrable (by Step 7) and it converges to $a (x, t) u (x, t)$ almost everywhere in $Ω_{T}$ , Vitali’s theorem (see [8] pag.122, n.4) implies $\begin{matrix} lim_{n \to \infty} \iint_{Ω_{T}} a_{n} (x, t) u_{n} φ = \iint_{Ω_{T}} a (x, t) u φ for every φ \in C^{\infty} (\overline{Ω_{T}}) \end{matrix}$ as we desired.

Putting together all the results in the previous steps, we can conclude that the function u satisfies $\begin{matrix} \iint_{Ω_{T}} {- u φ_{t} + M (x, t) \nabla u \cdot \nabla φ + a (x, t) u φ} = \iint_{Ω_{T}} f (x, t) φ + \int_{Ω} u_{0} φ (0), \end{matrix}$ for every $φ \in C^{\infty} (\overline{Ω_{T}})$ such that $φ (T) = 0$ , as we desired. Furthermore, the function u belongs to $L^{\infty} (0, T; L^{1} (Ω)) \cap L_{loc}^{\infty} (0, T; L^{\infty} (Ω)) \cap L_{loc}^{2} (0, T; H_{0}^{1} (Ω)) \cap L^{q} (0, T; W_{0}^{1, q} (Ω))$ for all $q < 2 - \frac{1}{N + 1}$ . □
Remark 3.1.
Notice that if $u_{0} \in L^{\infty} (Ω)$ it is possible to take $t_{0} = 0$ in (3.7). Hence, choosing also $k ⩾ max {Q, ‖ u_{0} ‖_{L^{\infty} (Ω)}}$ , by (3.7) (written with $t_{0} = 0$ and $t_{1} = t$ ) it follows $\begin{matrix} ‖ u ‖_{L^{\infty} (Ω_{T})} ⩽ max {Q, ‖ u_{0} ‖_{L^{\infty} (Ω)}} . \end{matrix}$ Hence, if the initial datum is bounded the solution constructed here is bounded in all the set $Ω_{T}$ and hence it belongs also to $L^{2} (0, T; H_{0}^{1} (Ω))$ .
Remark 3.2.
As noticed in the introduction, the presence of the lower order term $a (x, t) u$ together with the control condition (1.4) is the reason of this very strong regularization phenomenon. Different improvement of regularity (like for example higher integrability) due to the presence of different type of lower order terms can be found in [18].
Remark 3.3.
As noticed in the introduction, the structure assumption (1.4) is not a necessary condition to guarantee the existence of a weak solution of (1.1). As a matter of fact, under the structure assumptions (1.2), (1.3), if (1.5) holds true, then problem (1.1) has at least a weak solution u satisfying (3.1). Moreover, u can be obtained as limit of the approximating solutions $u_{n}$ of (3.3). To prove this result it is sufficient to observe that all the estimates proved in Steps 4–6 above remain true since we have not used assumption (1.4). In particular, the sequence $u_{n}$ is bounded in $L^{q} (0, T; W_{0}^{1, q} (Ω))$ and the sequence $a_{n} u_{n}$ is bounded in $L^{1} (Ω_{T})$ . Consequently ${(u_{n})}_{t}$ is bounded in $L^{q} (0, T; W^{- 1, q} (Ω)) + L^{1} (Ω_{T})$ . Hence, using again a classical result due to Simon (see [22]) it follows that (up to subsequences) $u_{n}$ strongly converges to u in $L^{1} (Ω_{T})$ and a.e. in $Ω_{T}$ . Then the assert can be proved proceeding exactly as in Steps 6-8 above.

4. The autonomous case

We prove here that in the autonomous case: $\begin{matrix} (4.1) & M (x, t) = M (x), a (x, t) = a (x) and f (x, t) = f (x), \end{matrix}$ the following result holds

Theorem 4.1.

Assume that ( 1.2 )–( 1.5 ) and ( 4.1 ) hold true. Then for every choice of the initial datum $u_{0}$ in $L^{1} (Ω)$ there exists a global solution $u (t)$ of ( 1.11 ), i.e. a solution of ( 1.1 ) in every set $Ω_{T}$ , for every $T > 0$ . Moreover, u satisfies the regularity properties $\begin{matrix} (4.2) & u \in L_{loc}^{q} ([0, + \infty); W_{0}^{1, q} (Ω)) \cap L_{loc}^{\infty} ([0, + \infty); L^{1} (Ω)), \end{matrix}$ where q is as in ( 3.1 ), and $\begin{matrix} (4.3) & u \in L_{loc}^{\infty} ((0, + \infty); L^{\infty} (Ω)) \cap L_{loc}^{2} ((0, + \infty); H_{0}^{1} (Ω)) . \end{matrix}$ Finally, the following estimate holds $\begin{matrix} (4.4) & {‖ u (t) - w ‖}_{L^{\infty} (Ω)} ⩽ C_{2} \frac{‖ u_{0} - w ‖_{L^{1} (Ω)}}{t^{\frac{N}{2}} e^{σ t}} a.e. t > 0, \end{matrix}$ where w is the unique solution in $L^{\infty} (Ω) \cap H_{0}^{1} (Ω)$ of the associated stationary problem ( 1.12 ), $C_{2}$ depends only on α and N, and $σ = c (α) | Ω |^{- \frac{2}{N}}$ . In particular, it results $\begin{matrix} lim_{t \to + \infty} {‖ u (t) - w ‖}_{L^{\infty} (Ω)} = 0 . \end{matrix}$

Remark 4.1.

When $a (x, t) \equiv 0$ and f is in $L_{loc}^{2} ([0, + \infty); H^{- 1} (Ω))$ the convergence (in the weaker space $L^{2} (Ω)$ ) of the solution u of (1.1) to the solution w of a “suitable near” stationary problem can be found in [15]. Further asymptotic results (still assuming $a (x, t) \equiv 0$ ) can be found in [19] and [20].

Proof of Theorem 4.1.

The proof proceeds in two steps. In the first one we prove the existence of a global solution verifying the stated regularity. In the second step we conclude the proof of the theorem showing that also estimate (4.4) holds true.

Step 1. To construct a global solution of (1.1) satisfying the assertion, let us consider a global approximating sequence $u_{n}$ constructed as follows. For every $T > 0$ arbitrarily fixed $u_{n}$ solves (3.3), i.e. $\begin{matrix} (4.5) & \{\begin{matrix} {(u_{n})}_{t} - div (M (x) \nabla u_{n}) + a_{n} (x) u_{n} = f_{n} (x) & in Ω_{T}, \\ u_{n} (x, t) = 0 & on Γ_{T}, \\ u_{n} (x, 0) = u_{0, n} (x) & in Ω, \end{matrix} \end{matrix}$ where $a_{n}$ and $f_{n}$ are bounded functions defined as follows $\begin{matrix} (4.6) & f_{n} (x) = \frac{f (x)}{1 + \frac{1}{n} | f (x) |}, a_{n} (x) = \frac{a (x)}{1 + \frac{Q}{n} | a (x) |}, \end{matrix}$ and, as before $u_{0, n} (x) \in C^{1} (\bar{Ω})$ satisfies $\begin{matrix} ‖ u_{0, n} ‖_{L^{1} (Ω)} ⩽ ‖ u_{0} ‖_{L^{1} (Ω)} u_{0, n} \to u_{0} in L^{1} (Ω) . \end{matrix}$ Notice that the sequence $u_{n} \in C_{loc} ([0, + \infty); L^{2} (Ω)) \cap L_{loc}^{2} ([0, + \infty); H_{0}^{1} (Ω))$ is well defined and uniquely determined since for every fixed $T > 0$ problem (4.5) admits an unique solution in $C ([0, T]; L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω))$ . We recall that the proof of Theorem 3.1 shows that in every $Ω_{T}$ there exists a subsequence of $u_{n}$ converging to a solution u of (1.1) having the regularity properties stated in Theorem 3.1. Since we do not know if the solution obtained in this way is unique, to construct a global solution we proceed in the following way. Let $T_{0} > 0$ arbitrarily fixed. Proceeding as in the proof of Theorem 3.1 we deduce that there exists a subsequence of $u_{n}$ , that we denote $u_{n_{1}}$ , that converges to a solution $u^{(1)}$ of our problem in $Ω \times [0, T_{0}]$ . We recall that every element of the sequence $u_{n_{1}}$ is a global solution. Hence, again proceeding as in the proof of Theorem 3.1 it follows that there exists a subsequence of $u_{n_{1}}$ , that we denote $u_{n_{2}}$ , that converges to a solution $u^{(2)}$ of our problem in $Ω \times [0, 2 T_{0}]$ . By construction, it results $u^{(1)} = u^{(2)}$ in $Ω \times [0, T_{0}]$ . Iterating this construction we obtain a sequence $u^{(n)}$ of solutions in $Ω \times [0, n T_{0}]$ such that $u^{(n)} = u^{(m)}$ in $Ω \times [0, min {n, m} T_{0}]$ . Now, the function u defined in $Ω \times [0, + \infty)$ as $u = u^{(n)}$ in $Ω \times [0, n T_{0}]$ for every integer $n ⩾ 1$ , is well defined and is a global solution satisfying the regularity properties (4.2) and (4.3).

Hence, to conclude the proof it remains to show that u satisfies also estimate (4.4). As said above we show it in the next step.

Step 2. Let u the global solution constructed in step 1 and w be the unique solution in $L^{\infty} (Ω) \cap H_{0}^{1} (Ω)$ of (1.12) (see [1]). We point out that it is sufficient to prove estimate (4.4) for a.e. $t \in (0, T)$ where $T > 0$ is arbitrarily fixed. To this aim we observe that w can be constructed as the $L^{1} (Ω)$ limit (and a.e. limit) of $w_{n} \in L^{\infty} (Ω) \cap H_{0}^{1} (Ω)$ , where $w_{n}$ are the solutions of the following problems $\begin{matrix} (4.7) & \{\begin{matrix} - div (M (x) \nabla w_{n}) + a_{n} (x) w_{n} = f_{n} (x) & in Ω, \\ w_{n} (x) = 0 & on \partial Ω \end{matrix} \end{matrix}$ where $a_{n}$ and $f_{n}$ are as in (4.6) (see again [1]). Notice that $w_{n}$ is also a global solution of the following evolution problem $\begin{matrix} (4.8) & \{\begin{matrix} {(w_{n})}_{t} - div (M (x) \nabla w_{n}) + a_{n} (x) w_{n} = f_{n} (x) & in Ω_{\infty}, \\ w_{n} (x) = 0 & on \partial Ω \times (0, + \infty) \\ w_{n} (x, 0) = w_{n} (x) & in Ω \end{matrix} \end{matrix}$

Thanks to the regularity of the solutions $u_{n}$ and $w_{n}$ it is possible to take as “test functions” in both the equations satisfied by $u_{n}$ and $w_{n}$ the function $G_{k} (u_{n} - w_{n})$ ( $k > 0$ ) and subtracting the equality obtained in this way (and using also the coercivity assumption (1.2)) we obtain for every $0 < t_{1} < t_{2} ⩽ T$ $\begin{array}{l} \frac{1}{2} \int_{Ω} {| G_{k} (u_{n} - w_{n}) |}^{2} (t_{2}) - \frac{1}{2} \int_{Ω} {| G_{k} (u_{n} - w_{n}) |}^{2} (t_{1}) \\ (4.9) & + α \int_{t_{1}}^{t_{2}} \int_{Ω} {| \nabla G_{k} (u_{n} - w_{n}) |}^{2} + \int_{t_{1}}^{t_{2}} \int_{Ω} a (x) (u_{n} - w_{n}) G_{k} (u_{n} - w_{n}) ⩽ 0 \end{array}$ Using the Sobolev inequality and recalling that the last integral in (4.9) is non negative we deduce that for every positive k and $0 < t_{1} < t_{2} ⩽ T$ it results $\begin{array}{l} \frac{1}{2} \int_{Ω} {| G_{k} (u_{n} - w_{n}) |}^{2} (t_{2}) - \frac{1}{2} \int_{Ω} {| G_{k} (u_{n} - w_{n}) |}^{2} (t_{1}) \\ (4.10) & + α c_{S} \int_{t_{1}}^{t_{2}} \int_{Ω} {‖ G_{k} (u_{n} - w_{n}) ‖}_{L^{2^{*}} (Ω)}^{2} ⩽ 0 \end{array}$ Moreover, using as a test function $φ = {1 - \frac{1}{{(1 + | G_{k} (u_{n} - w_{n}) |)}^{δ}}} sign (u_{n} - w_{n})$ ( $δ > 0$ ) in both the equation satisfied by $u_{n}$ and $w_{n}$ and subtracting the results and reasoning as in the proof of inequality (3.10) we get for every $0 ⩽ t_{0} < t ⩽ T$ and $k > 0$ $\begin{matrix} (4.11) & {‖ G_{k} (u_{n} - w_{n}) (t_{0}) ‖}_{L^{1} (Ω)} ⩽ {‖ G_{k} (u_{n} - w_{n}) (t_{0}) ‖}_{L^{1} (Ω)} . \end{matrix}$ Hence the assumptions of Theorem 2.2 are satisfied and we deduce that the following estimate holds true $\begin{matrix} {‖ u_{n} (t) - w_{n} (t) ‖}_{L^{\infty} (Ω)} ⩽ C_{2} \frac{‖ u_{0, n} - w_{n} ‖_{L^{1} (Ω)}}{t^{\frac{N}{2}} e^{σ t}} for every t \in (0, T), \end{matrix}$ where $C_{2}$ is a positive constant depending only on α and N and $σ = c (α) | Ω |^{- \frac{2}{N}}$ . By the previous estimate and recalling the construction of the global solution u done in the previous step it follows the assertion. □

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