On stabilization of solutions of higher order evolution inequalities

Abstract

We obtain sharp conditions guaranteeing that every non-negative weak solution of the inequality $\begin{matrix} \sum_{| α | = m} \partial^{α} a_{α} (x, t, u) - u_{t} ⩾ f (x, t) g (u) in R_{+}^{n + 1} = R^{n} \times (0, \infty), m, n ⩾ 1, \end{matrix}$ stabilizes to zero as $t \to \infty$ . These conditions generalize the well-known Keller–Osserman condition on the growth of the function g at infinity.

Keywords

Higher order evolution inequalities nonlinearity stabilization

1. Introduction

We study non-negative solutions of the inequality $\begin{matrix} (1.1) & \sum_{| α | = m} \partial^{α} a_{α} (x, t, u) - u_{t} ⩾ f (x, t) g (u) in R_{+}^{n + 1} = R^{n} \times (0, \infty), m, n ⩾ 1, \end{matrix}$ where $a_{α}$ are Caratheodory functions such that $\begin{matrix} (1.2) & | a_{α} (x, t, ζ) | ⩽ A ζ^{p}, | α | = m, \end{matrix}$ with some constants $A > 0$ and $p ⩾ 1$ for almost all $(x, t) \in R_{+}^{n + 1}$ and for all $ζ \in [0, \infty)$ . It is also assumed that f is a measurable function on the set $R_{+}^{n + 1}$ , $g (ζ) > 0$ for all $ζ > 0$ , and $\begin{matrix} (1.3) & G (ζ) = g (ζ^{1 / p}) \end{matrix}$ is a non-decreasing convex function on the closed interval $[0, \infty)$ . As is customary, by $α = (α_{1}, \dots, α_{n})$ we denote a multi-index with $| α | = α_{1} + \dots + α_{n}$ and $\partial^{α} = \partial^{| α |} / (\partial_{x_{1}}^{α_{1}} \dots \partial_{x_{n}}^{α_{n}})$ , $x = (x_{1}, \dots, x_{n})$ .

Definition 1.1.
A non-negative function $u \in L_{p, loc} (R_{+}^{n + 1})$ is called a weak solution of (1.1) if $f (x, t) g (u) \in L_{1, loc} (R_{+}^{n + 1})$ and, moreover, for any non-negative function $φ \in C_{0}^{\infty} (R_{+}^{n + 1})$ the following inequality holds: $\begin{matrix} (1.4) & \int_{R_{+}^{n + 1}} \sum_{| α | = m} {(- 1)}^{m} a_{α} (x, t, u) \partial^{α} φ d x d t + \int_{R_{+}^{n + 1}} u φ_{t} d x d t ⩾ \int_{R_{+}^{n + 1}} f (x, t) g (u) φ d x d t . \end{matrix}$

Questions treated in this paper were earlier investigated mainly for differential operators of the second order [1,3–6,8–11,13,14,16,17,19]. The case of higher order operators has been studied much less [2,7]. Our aim is to obtain sufficient stabilization conditions for weak solution of inequality (1.1). In so doing, no initial conditions on solutions of (1.1) are imposed. We even admit that $u (x, t)$ can tend to infinity as $t \to + 0$ . We also impose no ellipticity conditions on the coefficients $a_{α}$ of the differential operator. Thus, our results can be applied to both parabolic and so-called anti-parabolic inequalities.
2. Main results

Theorem 2.1.
Let $\begin{matrix} (2.1) & \int_{1}^{\infty} g^{- 1 / m} (ζ) ζ^{p / m - 1} d ζ < \infty \end{matrix}$ and $\begin{matrix} (2.2) & lim_{t \to \infty} \underset{K \times (t, \infty)}{ess inf} f = \infty \end{matrix}$ for any compact set $K \subset R^{n}$ . Then every non-negative weak solution of ( 1.1 ) stabilizes to zero as $t \to \infty$ in the $L_{1}$ norm on an arbitrary compact set $K \subset R^{n}$ , i.e. $\begin{matrix} (2.3) & lim_{t \to \infty} \underset{τ \in (t, \infty)}{ess sup} {‖ u (\cdot, τ) ‖}_{L_{1} (K)} = 0 . \end{matrix}$

The proof of Theorem 2.1 is given in Section 3.
Remark 2.1.
Since $u \in L_{1, loc} (R_{+}^{n + 1})$ , the norm on the left in (2.3) is defined for almost all $τ \in (0, \infty)$ .
Example 2.1.
Consider the inequality $\begin{matrix} (2.4) & Δ^{m / 2} u^{p} - u_{t} ⩾ f (x, t) u^{λ} in R_{+}^{n + 1}, \end{matrix}$ where m is a positive even integer and λ is a real number. By Theorem 2.1, if $\begin{matrix} (2.5) & λ > p \end{matrix}$ and (2.2) is valid, then every non-negative weak solution of (2.4) stabilizes to zero as $t \to \infty$ in $L_{1}$ norm on an arbitrary compact subset of $R^{n}$ .

Now, let us consider the inequality $\begin{matrix} (2.6) & Δ^{m / 2} u^{p} - u_{t} ⩾ f (x, t) u^{p} {ln}^{ν} (1 + u) in R_{+}^{n + 1}, \end{matrix}$ where ν is a real number. In other words, we examine the case of the critical exponent $λ = p$ in the right-hand side of (2.4) “spoiled” by the logarithm. As before, we assume that m is a positive even integer.

It can easily be seen that (2.1) is equivalent to the condition $\begin{matrix} (2.7) & ν > m . \end{matrix}$ Thus, if (2.2) and (2.7) are valid, then Theorem 2.1 implies that every non-negative weak solution of (2.6) stabilizes to zero as $t \to \infty$ in $L_{1}$ norm on an arbitrary compact subset of $R^{n}$ .

Condition (2.7) is the best possible. Indeed, let us show that there exists a positive function $f \in C^{\infty} (R_{+}^{n + 1})$ for which (2.2) holds and the inequality $\begin{matrix} (2.8) & Δ^{m / 2} u^{p} - u_{t} ⩾ f (x, t) u^{p} {ln}^{m} (1 + u) in R_{+}^{n + 1} \end{matrix}$ has a classical solution satisfying the bound $u (x, t) ⩾ e$ for all $(x, t) \in R_{+}^{n + 1}$ . It is obvious that this solution is also a solution of (2.6) for all $ν ⩽ m$ . We shall seek it in the form $\begin{matrix} (2.9) & u (x, t) = e^{e^{w (x, t)}}, \end{matrix}$ where $\begin{matrix} w (x, t) = (t + 1) \sum_{i = 1}^{n} e^{x_{i}} . \end{matrix}$ By direct differentiation, one can verify that $\begin{matrix} Δ^{m / 2} u^{p} ⩾ \sum_{i = 1}^{n} p^{m} {(t + 1)}^{m} e^{m x_{i}} e^{m w (x, t)} u^{p} + \sum_{i = 1}^{n} p (t + 1) e^{x_{i}} e^{w (x, t)} u^{p} \end{matrix}$ for all $(x, t) \in R_{+}^{n + 1}$ . Since $\begin{matrix} u_{t} = \sum_{i = 1}^{n} e^{x_{i}} e^{w (x, t)} u \end{matrix}$ and $p ⩾ 1$ , this yields $\begin{matrix} (2.10) & Δ^{m / 2} u^{p} - u_{t} ⩾ \sum_{i = 1}^{n} p^{m} {(t + 1)}^{m} e^{m x_{i}} e^{m w (x, t)} u^{p} \end{matrix}$ for all $(x, t) \in R_{+}^{n + 1}$ . It can be seen that $\begin{matrix} e^{w (x, t)} = ln u ⩾ \frac{1}{2} ln (1 + u) \end{matrix}$ for all $(x, t) \in R_{+}^{n + 1}$ . Thus, (2.10) implies inequality (2.8) with $\begin{matrix} f (x, t) = \frac{1}{2^{m}} \sum_{i = 1}^{n} p^{m} {(t + 1)}^{m} e^{m x_{i}} . \end{matrix}$

Note that, along with (2.7), we have established the exactness of condition (2.5). In fact, any solution of (2.8) satisfying the inequality $u ⩾ e$ on the set $R_{+}^{n + 1}$ is also a solution of (2.4) for all $λ ⩽ p$ .
Remark 2.2.
If $m = 2$ and $p = 1$ , then (2.1) takes the form $\begin{matrix} (2.11) & \int_{1}^{\infty} {(g (t) t)}^{- 1 / 2} d t < \infty . \end{matrix}$ It is easy to see that (2.11) is equivalent to the well-known Keller–Osserman condition $\begin{matrix} (2.12) & \int_{1}^{\infty} {(\int_{1}^{t} g (s) d s)}^{- 1 / 2} d t < \infty \end{matrix}$ on the growth of the function g at infinity [12,18] which plays an important role in the theory of semilinear elliptic and parabolic equations (see, for instance, [19] and references therein). Indeed, we have $\begin{matrix} \int_{1}^{t} g (s) d s ⩾ \int_{t / 2}^{t} g (s) d s ⩾ \frac{t}{2} g (\frac{t}{2}), t > 2, \end{matrix}$ since g is a non-decreasing positive function on the interval $(0, \infty)$ . Hence, (2.11) implies (2.12). On the other hand, $\begin{matrix} \int_{1}^{t} g (s) d s ⩽ t g (t), t > 1; \end{matrix}$ therefore, (2.11) follows from (2.12).

We can in this context call (2.1) as a generalized Keller–Osserman condition.
Example 2.2.
Consider the first-order differential inequality $\begin{matrix} (2.13) & \sum_{i = 1}^{n} \frac{\partial u^{p}}{\partial x_{i}} - u_{t} ⩾ f (x, t) u^{λ} in R_{+}^{n + 1} . \end{matrix}$ As in the case of (2.4), it can be seen that (2.1) is equivalent to (2.5). Thus, according to Theorem 2.1, if (2.2) and (2.5) are valid, then every non-negative weak solution of (2.13) stabilizes to zero as $t \to \infty$ in $L_{1}$ norm on an arbitrary compact subset of $R^{n}$ .

Now, we consider the inequality $\begin{matrix} (2.14) & \sum_{i = 1}^{n} \frac{\partial u^{p}}{\partial x_{i}} - u_{t} ⩾ f (x, t) u^{p} {ln}^{ν} (1 + u) in R_{+}^{n + 1} . \end{matrix}$ For $ν = 0$ , we obviously obtain (2.13) with the critical exponent $λ = p$ . By Theorem 2.1, if (2.2) holds and $\begin{matrix} (2.15) & ν > 1, \end{matrix}$ then every non-negative weak solution of (2.14) stabilizes to zero as $t \to \infty$ in $L_{1}$ norm on an arbitrary compact subset of $R^{n}$ . Condition (2.15) is the best possible. To see this, it is sufficient to verify that (2.9) is a solution of the inequality $\begin{matrix} (2.16) & \sum_{i = 1}^{n} \frac{\partial u^{p}}{\partial x_{i}} - u_{t} ⩾ f (x, t) u^{p} ln (1 + u) in R_{+}^{n + 1}, \end{matrix}$ where $\begin{matrix} f (x, t) = \frac{1}{2} \sum_{i = 1}^{n} p t e^{x_{i}} . \end{matrix}$

Since any solution of (2.16) satisfying the bound $u ⩾ e$ on the set $R_{+}^{n + 1}$ is a solution of (2.13) for all $λ ⩽ p$ , we also establish the exactness of condition (2.5) for stabilization of solutions of inequality (2.13).
Theorem 2.2.
Under the hypotheses of Theorem 2.1 , let $g (0) > 0$ . Then inequality ( 1.1 ) has no non-negative weak solutions.

The proof of Theorem 2.2 is given in Section 3. Remark 2.3.
In Theorem 2.2, the condition $g (0) > 0$ is essential. Indeed, in the case of $g (0) = 0$ , we have $g (ζ) ⩽ g (1) ζ$ for all $0 ⩽ ζ ⩽ 1$ since g is a convex function.1
¹
The function g is obviously convex since the function G defined by (1.3) is convex and, moreover, $p ⩾ 1$ .

Thus, putting $\begin{matrix} u (x, t) = e^{- k t^{2}}, \end{matrix}$ where $k > 0$ is a sufficiently large real number, we obtain $\begin{matrix} L u^{p} - u_{t} ⩾ t g (u) in R_{+}^{n + 1} \end{matrix}$ for any linear differential operator $L$ with constant coefficients not containing derivatives with respect to the variable t.

3. Proof of Theorems 2.1 and 2.2

Below, it is assumed that u is a non-negative weak solution of (1.1). By C we mean various positive constants that can depend only on m, n, and p.

Let us use the following notations. We denote $B_{r} = {x \in R^{n} : | x | < r}$ and $Q_{r}^{t_{1}, t_{2}} = {(x, t) \in R^{n + 1} : | x | < r, t_{1} < t < t_{2}}$ . Further, let $ω \in C^{\infty} (R)$ be a non-negative function such that $supp ω \subset (- 1, 1)$ and $\begin{matrix} \int_{- \infty}^{\infty} ω d t = 1 . \end{matrix}$ We need the Steklov–Schwartz averaging kernel $\begin{matrix} (3.1) & ω_{h} (t) = \frac{1}{h} w (\frac{t}{h}), h > 0 . \end{matrix}$

Lemma 3.1.
Let $0 < r_{1} < r_{2}$ and $0 < h < τ_{1} < τ_{2} < τ$ be some real numbers with $r_{2} ⩽ 2 r_{1}$ . If $f (x, t) ⩾ 0$ for almost all $(x, t) \in Q_{r_{2}}^{τ - τ_{2}, τ + h}$ , then $\begin{array}{l} \frac{1}{{(r_{2} - r_{1})}^{m}} \int_{Q_{r_{2}}^{τ - τ_{2}, τ + h} ∖ Q_{r_{1}}^{τ - τ_{2}, τ + h}} u^{p} d x d t + \frac{1}{τ_{2} - τ_{1}} \int_{Q_{r_{2}}^{τ - τ_{2}, τ - τ_{1}}} u d x d t \\ (3.2) & ⩾ C (\int_{Q_{r_{1}}^{τ - τ_{1}, τ - h}} f (x, t) g (u) d x d t + \int_{Q_{r_{1}}^{τ - h, τ + h}} ω_{h} (τ - t) u d x d t) . \end{array}$
Proof.
We take a non-decreasing function $φ_{0} \in C^{\infty} (R)$ such that $\begin{matrix} φ_{0} |_{(- \infty, 0]} = 0 and φ_{0} |_{[1, \infty)} = 1 . \end{matrix}$ Also let $\begin{matrix} η (t) = \int_{t}^{\infty} ω_{h} (τ - ξ) d ξ . \end{matrix}$ Using $\begin{matrix} φ (x, t) = φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) φ_{0} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) η (t) \end{matrix}$ as a test function in (1.4), we obtain $\begin{array}{l} \int_{R_{+}^{n + 1}} \sum_{| α | = m} {(- 1)}^{m} a_{α} (x, t, u) \partial^{α} φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) φ_{0} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) η (t) d x d t \\ + \int_{R_{+}^{n + 1}} u φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) \frac{\partial φ_{0}}{\partial t} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) η (t) d x d t \\ ⩾ \int_{R_{+}^{n + 1}} f (x, t) g (u) φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) φ_{0} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) η (t) d x d t \\ (3.3) & + \int_{R_{+}^{n + 1}} u φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) φ_{0} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) ω_{h} (τ - t) d x d t . \end{array}$ Condition (1.2) allows us to assert that $\begin{array}{l} \frac{1}{{(r_{2} - r_{1})}^{m}} \int_{Q_{r_{2}}^{τ - τ_{2}, τ + h} ∖ Q_{r_{1}}^{τ - τ_{2}, τ + h}} u^{p} d x d t \\ ⩾ C \int_{R_{+}^{n + 1}} \sum_{| α | = m} | a_{α} (x, t, u) \partial^{α} φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) φ_{0} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) η (t) | d x d t . \end{array}$ In so doing, we obviously have $\begin{matrix} \frac{1}{τ_{2} - τ_{1}} \int_{Q_{r_{2}}^{τ - τ_{2}, τ - τ_{1}}} u d x d t ⩾ C \int_{R_{+}^{n + 1}} u φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) \frac{\partial φ_{0}}{\partial t} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) η (t) d x d t \end{matrix}$ and $\begin{matrix} \int_{R_{+}^{n + 1}} f (x, t) g (u) φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) φ_{0} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) η (t) d x d t ⩾ \int_{Q_{r_{1}}^{τ - τ_{1}, τ - h}} f (x, t) g (u) d x d t . \end{matrix}$ Finally, since $ω_{h} (τ - t) = 0$ for all $t \notin (τ - h, τ + h)$ and $φ_{0} ((t - τ + τ_{2}) / (τ_{2} - τ_{1})) = 1$ for all $t > τ - h$ , the second summand in the right-hand side of (3.3) can be estimated as follows: $\begin{matrix} \int_{R_{+}^{n + 1}} u φ_{0} (\frac{r_{2} - | x |}{r_{2} - r_{1}}) φ_{0} (\frac{t - τ + τ_{2}}{τ_{2} - τ_{1}}) ω_{h} (τ - t) d x d t ⩾ \int_{Q_{r_{1}}^{τ - h, τ + h}} ω_{h} (τ - t) u d x d t . \end{matrix}$ Thus, combining (3.3) with the last four inequalities, we deduce (3.2). □

For any real number $R > 0$ we define the function $\begin{matrix} J_{R} (r, τ) = \frac{1}{meas Q_{2 R}^{τ - 2^{m} R^{m}, τ}} \int_{Q_{r}^{τ - r^{m}, τ}} u^{p} d x d t, \end{matrix}$ where $0 < r ⩽ 2 R$ and $τ > 2^{m} R^{m}$ .
Lemma 3.2.
Let $R > 0$ and $τ > 0$ be some real numbers such that $τ > 2^{m} R^{m}$ and $f (x, t) ⩾ 0$ for almost all $(x, t) \in Q_{2 R}^{τ - 2^{m} R^{m}, \infty}$ . Then for all $r_{1}$ and $r_{2}$ satisfying the condition $R ⩽ r_{1} < r_{2} ⩽ 2 R$ at least one of the following two inequalities is valid: $\begin{array}{l} (3.4) & J_{R} (r_{2}, τ) - J_{R} (r_{1}, τ) ⩾ C {(r_{2} - r_{1})}^{m} G (J_{R} (r_{1}, τ)) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f, \\ (3.5) & J_{R} (r_{2}, τ) - J_{R} (r_{1}, τ) ⩾ C R^{m p - 1} (r_{2} - r_{1}) G^{p} (J_{R} (r_{1}, τ)) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{p}, \end{array}$ where the function G is defined by ( 1.3 ).
Proof.
Inequality (3.2) of Lemma 3.1 with $τ_{1} = r_{1}^{m}$ and $τ_{2} = r_{2}^{m}$ yields $\begin{array}{l} \frac{1}{{(r_{2} - r_{1})}^{m}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ + h} ∖ Q_{r_{1}}^{τ - r_{2}^{m}, τ + h}} u^{p} d x d t + \frac{1}{r_{2}^{m} - r_{1}^{m}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u d x d t \\ ⩾ C \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ - h}} f (x, t) g (u) d x d t \end{array}$ for all sufficiently small $h > 0$ . Note that the second summand in the right-hand side of (3.2) is non-negative; therefore, it can be dropped. Passing to the limit as $h \to + 0$ in the last estimate, we obviously obtain $\begin{array}{l} \frac{1}{{(r_{2} - r_{1})}^{m}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{2}^{m}, τ}} u^{p} d x d t + \frac{1}{r_{2}^{m} - r_{1}^{m}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u d x d t \\ (3.6) & ⩾ C \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} f (x, t) g (u) d x d t . \end{array}$

Let us show that $\begin{matrix} (3.7) & \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} f (x, t) g (u) d x d t ⩾ G (J_{R} (r_{1}, τ)) meas Q_{r_{1}}^{τ - r_{1}^{m}, τ} \underset{Q_{r_{1}}^{τ - r_{1}^{m}, τ}}{ess inf} f . \end{matrix}$ Indeed, if $\begin{matrix} \underset{Q_{r_{1}}^{τ - r_{1}^{m}, τ}}{ess inf} f = 0, \end{matrix}$ then (3.7) is evident; therefore, it can be assumed without loss of generality that $\begin{matrix} \underset{Q_{r_{1}}^{τ - r_{1}^{m}, τ}}{ess inf} f > 0 . \end{matrix}$ In this case, taking into account the inequality $\begin{matrix} \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} f (x, t) g (u) d x d t < \infty, \end{matrix}$ we readily obtain $\begin{matrix} \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} g (u) d x d t < \infty \end{matrix}$ and $\begin{matrix} (3.8) & \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} f (x, t) g (u) d x d t ⩾ \underset{Q_{r_{1}}^{τ - r_{1}^{m}, τ}}{ess inf} f \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} g (u) d x d t . \end{matrix}$ Since G is a non-decreasing convex function, we also have $\begin{array}{l} \frac{1}{meas Q_{r_{1}}^{τ - r_{1}^{m}, τ}} \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} g (u) d x d t & = \frac{1}{meas Q_{r_{1}}^{τ - r_{1}^{m}, τ}} \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} G (u^{p}) d x d t \\ ⩾ G (\frac{1}{meas Q_{r_{1}}^{τ - r_{1}^{m}, τ}} \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} u^{p} d x d t) ⩾ G (J_{R} (r_{1}, τ)), \end{array}$ whence in turn it follows that $\begin{matrix} \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} g (u) d x d t ⩾ G (J_{R} (r_{1}, τ)) meas Q_{r_{1}}^{τ - r_{1}^{m}, τ} . \end{matrix}$ In view of (3.8), this yields (3.7).

Assume that $\begin{matrix} (3.9) & \frac{1}{{(r_{2} - r_{1})}^{m}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{2}^{m}, τ}} u^{p} d x d t ⩾ \frac{1}{r_{2}^{m} - r_{1}^{m}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u d x d t . \end{matrix}$ Then (3.6) implies the inequality $\begin{matrix} (3.10) & \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{2}^{m}, τ}} u^{p} d x d t ⩾ C {(r_{2} - r_{1})}^{m} \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} f (x, t) g (u) d x d t . \end{matrix}$ We obviously have $\begin{matrix} Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{2}^{m}, τ} \subset Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{1}^{m}, τ}; \end{matrix}$ therefore, $\begin{array}{l} J_{R} (r_{2}, τ) - J_{R} (r_{1}, τ) & = \frac{1}{meas Q_{2 R}^{τ - 2^{m} R^{m}, τ}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{1}^{m}, τ}} u^{p} d x d t \\ ⩾ \frac{1}{meas Q_{2 R}^{τ - 2^{m} R^{m}, τ}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{2}^{m}, τ}} u^{p} d x d t . \end{array}$ Combining this with (3.10), we obtain $\begin{matrix} (3.11) & J_{R} (r_{2}, τ) - J_{R} (r_{1}, τ) ⩾ \frac{C {(r_{2} - r_{1})}^{m}}{meas Q_{2 R}^{τ - 2^{m} R^{m}, τ}} \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} f (x, t) g (u) d x d t . \end{matrix}$ Further, it does not present any particular problem to verify that $\begin{matrix} (3.12) & A_{1} R^{m + n} ⩽ meas Q_{r}^{τ - r^{m}, τ} ⩽ A_{2} R^{m + n} \end{matrix}$ for all $R ⩽ r ⩽ 2 R$ , where the constants $A_{1} > 0$ and $A_{2} > 0$ depend only on m and n. Thus, combining (3.7) and (3.11), we arrive at (3.4).

Now, assume that the opposite inequality to (3.9) holds, i.e. $\begin{matrix} \frac{1}{{(r_{2} - r_{1})}^{m}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{2}^{m}, τ}} u^{p} d x d t < \frac{1}{r_{2}^{m} - r_{1}^{m}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u d x d t . \end{matrix}$ Combining this with (3.6), we have $\begin{matrix} (3.13) & \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u d x d t ⩾ C (r_{2}^{m} - r_{1}^{m}) \int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} f (x, t) g (u) d x d t . \end{matrix}$ By the Hölder inequality, one can show that $\begin{matrix} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u d x d t ⩽ {(meas Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}})}^{(p - 1) / p} {(\int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u^{p} d x d t)}^{1 / p}, \end{matrix}$ whence, taking into account the fact that $\begin{matrix} meas Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}} ⩽ C (r_{2}^{m} - r_{1}^{m}) R^{n}, \end{matrix}$ we obtain $\begin{matrix} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u d x d t ⩽ C {(r_{2}^{m} - r_{1}^{m})}^{(p - 1) / p} R^{n (p - 1) / p} {(\int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u^{p} d x d t)}^{1 / p} . \end{matrix}$ Thus, (3.13) implies the estimate $\begin{matrix} (3.14) & \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u^{p} d x d t ⩾ \frac{C (r_{2}^{m} - r_{1}^{m})}{R^{n (p - 1)}} {(\int_{Q_{r_{1}}^{τ - r_{1}^{m}, τ}} f (x, t) g (u) d x d t)}^{p} . \end{matrix}$ Due to the obvious inequality $\begin{matrix} r_{2}^{m} - r_{1}^{m} ⩾ C (r_{2} - r_{1}) R^{m - 1} \end{matrix}$ and relationship (3.7) estimate (3.14) yields $\begin{matrix} (3.15) & \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u^{p} d x d t ⩾ C R^{m (p + 1) + n - 1} (r_{2} - r_{1}) G^{p} (J_{R} (r_{1}, τ)) \underset{Q_{r_{1}}^{τ - r_{1}^{m}, τ}}{ess inf} f^{p} . \end{matrix}$ Since $\begin{matrix} Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}} \subset Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{1}^{m}, τ}, \end{matrix}$ we have $\begin{array}{l} J_{R} (r_{2}, τ) - J_{R} (r_{1}, τ) & = \frac{1}{meas Q_{2 R}^{τ - 2^{m} R^{m}, τ}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ} ∖ Q_{r_{1}}^{τ - r_{1}^{m}, τ}} u^{p} d x d t \\ ⩾ \frac{1}{meas Q_{2 R}^{τ - 2^{m} R^{m}, τ}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u^{p} d x d t . \end{array}$ In view of (3.12), this implies the estimate $\begin{matrix} J_{R} (r_{2}, τ) - J_{R} (r_{1}, τ) ⩾ \frac{C}{R^{m + n}} \int_{Q_{r_{2}}^{τ - r_{2}^{m}, τ - r_{1}^{m}}} u^{p} d x d t \end{matrix}$ from which, taking into account (3.15), we obtain (3.5). □
Lemma 3.3.
Let $R > 0$ and $τ > 0$ be some real numbers such that $τ > 2^{m} R^{m}$ and $\begin{matrix} (3.16) & \int_{Q_{R}^{τ - R^{m}, τ}} u^{p} d x d t > 0 . \end{matrix}$ If $f (x, t) ⩾ 0$ for almost all $(x, t) \in Q_{2 R}^{τ - 2^{m} R^{m}, \infty}$ , then $\begin{array}{l} \int_{J_{R} (R, τ)}^{\infty} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ + \int_{J_{R} (R, τ)}^{\infty} \frac{d ζ}{G (ζ / 2)} + \int_{J_{R} (R, τ)}^{\infty} \frac{d ζ}{G^{p} (ζ / 2)} \\ (3.17) & ⩾ C min {R \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{1 / m}, R^{m} \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f, R^{m p} \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{p}}, \end{array}$ where the function G is defined by ( 1.3 ).
Proof.
We construct a finite sequence of real numbers ${r_{i}}_{i = 0}^{l}$ as follows. Let us take $r_{0} = R$ . Assume further that $r_{i}$ is already defined. If $r_{i} ⩾ 3 R / 2$ , then we put $l = i$ and stop; otherwise we take $\begin{matrix} r_{i + 1} = sup {r \in [r_{i}, 2 R] : J_{R} (r, τ) ⩽ 2 J_{R} (r_{i}, τ)} . \end{matrix}$ Since $u \in L_{p, loc} (R_{+}^{n + 1})$ , this procedure must terminate at a finite step. It follows from (3.16) that $J_{R} (r_{i}, τ) > 0$ for all $0 ⩽ i ⩽ l$ ; therefore, ${r_{i}}_{i = 0}^{l}$ is a strictly increasing sequence. It can also be seen that $\begin{matrix} J_{R} (r_{i + 1}, τ) = 2 J_{R} (r_{i}, τ) for all 0 ⩽ i ⩽ l - 2 \end{matrix}$ and $\begin{matrix} J_{R} (r_{l}, τ) ⩽ 2 J_{R} (r_{l - 1}, τ) . \end{matrix}$ Moreover, if $J_{R} (r_{l}, τ) < 2 J_{R} (r_{l - 1}, τ)$ , then $r_{l} = 2 R$ and $r_{l} - r_{l - 1} ⩾ R / 2$ .

In view of Lemma 3.2, for any $0 ⩽ i ⩽ l - 1$ at least one of the following two inequalities is valid: $\begin{array}{l} (3.18) & J_{R} (r_{i + 1}, τ) - J_{R} (r_{i}, τ) ⩾ C {(r_{i + 1} - r_{i})}^{m} G (J_{R} (r_{i}, τ)) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f, \\ (3.19) & J_{R} (r_{i + 1}, τ) - J_{R} (r_{i}, τ) ⩾ C R^{m p - 1} (r_{i + 1} - r_{i}) G^{p} (J_{R} (r_{i}, τ)) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{p} . \end{array}$

By $Ξ_{1}$ we denote the set of integers $0 ⩽ i ⩽ l - 1$ for which (3.18) is valid. In so doing, let $Ξ_{2} = {0, \dots, l - 1} ∖ Ξ_{1}$ .

We claim that $\begin{array}{l} \int_{J_{R} (r_{i}, τ)}^{J_{R} (r_{i + 1}, τ)} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ + \int_{J_{R} (r_{i}, τ)}^{J_{R} (r_{i + 1}, τ)} \frac{d ζ}{G (ζ / 2)} \\ (3.20) & ⩾ C (r_{i + 1} - r_{i}) min {\underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{1 / m}, R^{m - 1} \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f} \end{array}$ for all $i \in Ξ_{1}$ . Indeed, (3.18) implies the inequality $\begin{matrix} {(\frac{J_{R} (r_{i + 1}, τ) - J_{R} (r_{i}, τ)}{G (J_{R} (r_{i}, τ))})}^{1 / m} ⩾ C (r_{i + 1} - r_{i}) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{1 / m} . \end{matrix}$ If $J_{R} (r_{i + 1}, τ) = 2 J_{R} (r_{i}, τ)$ , then due to monotonicity of the function G we have $\begin{matrix} \int_{J_{R} (r_{i}, τ)}^{J_{R} (r_{i + 1}, τ)} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ ⩾ C {(\frac{J_{R} (r_{i + 1}, τ) - J_{R} (r_{i}, τ)}{G (J_{R} (r_{i}, τ))})}^{1 / m} . \end{matrix}$ Combining the last two inequalities, we get $\begin{matrix} \int_{J_{R} (r_{i}, τ)}^{J_{R} (r_{i + 1}, τ)} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ ⩾ C (r_{i + 1} - r_{i}) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{1 / m}, \end{matrix}$ whence (3.20) follows at once.

On the other hand, if $J_{R} (r_{i + 1}, τ) < 2 J_{R} (r_{i}, τ)$ for some $i \in Ξ_{1}$ , then $i = l - 1$ and, moreover, $r_{i + 1} - r_{i} ⩾ R / 2$ . Thus, (3.18) implies the estimate $\begin{matrix} \frac{J_{R} (r_{i + 1}, τ) - J_{R} (r_{i}, τ)}{G (J_{R} (r_{i}, τ))} ⩾ C R^{m - 1} (r_{i + 1} - r_{i}) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f . \end{matrix}$ Since $\begin{matrix} \int_{J_{R} (r_{i}, τ)}^{J_{R} (r_{i + 1}, τ)} \frac{d ζ}{G (ζ / 2)} ⩾ \frac{C (J_{R} (r_{i + 1}, τ) - J_{R} (r_{i}, τ))}{G (J_{R} (r_{i}, τ))}, \end{matrix}$ this yields $\begin{matrix} \int_{J_{R} (r_{i}, τ)}^{J_{R} (r_{i + 1}, τ)} \frac{d ζ}{G (ζ / 2)} ⩾ C R^{m - 1} (r_{i + 1} - r_{i}) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f, \end{matrix}$ whence we again derive (3.20).

In a similar way, it can be shown that $\begin{matrix} (3.21) & \int_{J_{R} (r_{i}, τ)}^{J_{R} (r_{i + 1}, τ)} \frac{d ζ}{G^{p} (ζ / 2)} ⩾ C R^{m p - 1} (r_{i + 1} - r_{i}) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{p} \end{matrix}$ for all $i \in Ξ_{2}$ . Indeed, taking into account (3.19), we have $\begin{matrix} \frac{J_{R} (r_{i + 1}, τ) - J_{R} (r_{i}, τ)}{G^{p} (J_{R} (r_{i}, τ))} ⩾ C R^{m p - 1} (r_{i + 1} - r_{i}) \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{p} . \end{matrix}$ In view of the inequality $\begin{matrix} \int_{J_{R} (r_{i}, τ)}^{J_{R} (r_{i + 1}, τ)} \frac{d ζ}{G^{p} (ζ / 2)} ⩾ \frac{C (J_{R} (r_{i + 1}, τ) - J_{R} (r_{i}, τ))}{G^{p} (J_{R} (r_{i}, τ))}, \end{matrix}$ this obviously implies (3.21).

Further, summing (3.20) over all $i \in Ξ_{1}$ , we obtain $\begin{array}{l} \int_{J_{R} (r_{0}, τ)}^{\infty} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ + \int_{J_{R} (r_{0}, τ)}^{\infty} \frac{d ζ}{G (ζ / 2)} \\ ⩾ C min {\underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{1 / m}, R^{m - 1} \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f} \sum_{i \in Ξ_{1}} (r_{i + 1} - r_{i}) . \end{array}$ Analogously, (3.21) yields $\begin{matrix} \int_{J_{R} (r_{0}, τ)}^{\infty} \frac{d ζ}{G^{p} (ζ / 2)} ⩾ C R^{m p - 1} \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{p} \sum_{i \in Ξ_{2}} (r_{i + 1} - r_{i}) . \end{matrix}$ Thus, summing the last two inequalities, we conclude that $\begin{array}{l} \int_{J_{R} (r_{0}, τ)}^{\infty} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ + \int_{J_{R} (r_{0}, τ)}^{\infty} \frac{d ζ}{G (ζ / 2)} + \int_{J_{R} (r_{0}, τ)}^{\infty} \frac{d ζ}{G^{p} (ζ / 2)} \\ ⩾ C min {\underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{1 / m}, R^{m - 1} \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f, R^{m p - 1} \underset{Q_{2 R}^{τ - 2^{m} R^{m}, τ}}{ess inf} f^{p}} (r_{l} - r_{0}) . \end{array}$

To complete the proof, it remains to note that $r_{l} - r_{0} ⩾ R / 2$ by the construction of the sequence ${r_{i}}_{i = 0}^{l}$ . □

We need the following known assertion.
Lemma 3.4 (see [15, Lemma 2.3]).

Let $ψ : (0, \infty) \to (0, \infty)$ and $γ : (0, \infty) \to (0, \infty)$ be measurable functions satisfying the condition $\begin{matrix} γ (ζ) ⩽ \underset{(ζ / θ, θ ζ)}{ess inf} ψ \end{matrix}$ with some real number $θ > 1$ for almost all $ζ \in (0, \infty)$ . Also assume that $0 < α ⩽ 1$ , $M_{1} > 0$ , $M_{2} > 0$ , and $ν > 1$ are some real numbers with $M_{2} ⩾ ν M_{1}$ . Then $\begin{matrix} {(\int_{M_{1}}^{M_{2}} γ^{- α} (ζ) ζ^{α - 1} d ζ)}^{1 / α} ⩾ A \int_{M_{1}}^{M_{2}} \frac{d ζ}{ψ (ζ)}, \end{matrix}$ where the constant $A > 0$ depends only on α, ν, and θ.

Lemma 3.5.

Let the hypotheses of Theorem 2.1 be valid, then $\begin{matrix} (3.22) & \int_{Q_{R}^{τ - R^{m}, τ}} u^{p} d x d t \to 0 as τ \to \infty \end{matrix}$ for any real number $R > 0$ .

Proof.

Condition (2.1) is equivalent to $\begin{matrix} (3.23) & \int_{1}^{\infty} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ < \infty, \end{matrix}$ where the function G is defined by (1.3). To verify this, it suffices to make the replacement $ζ = {(ξ / 2)}^{1 / p}$ in formula (2.1). By Lemma 3.4, we obtain $\begin{matrix} {(\int_{1}^{\infty} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ)}^{m} ⩾ C \int_{1}^{\infty} \frac{d ζ}{G (ζ)} . \end{matrix}$ This allows us to assert that $\begin{matrix} (3.24) & \int_{1}^{\infty} \frac{d ζ}{G (ζ)} < \infty . \end{matrix}$ Since G is a non-decreasing positive function on the interval $(0, \infty)$ , we have $G^{p} (ζ) ⩾ G^{p - 1} (1) G (ζ)$ for all $ζ ⩾ 1$ . Hence, we can also assert that $\begin{matrix} (3.25) & \int_{1}^{\infty} \frac{d ζ}{G^{p} (ζ)} < \infty . \end{matrix}$

Assume that $R > 0$ is some given real number. In view of (2.2), f is a non-negative function almost everywhere on $Q_{2 R}^{τ - 2^{m} R^{m}, \infty}$ for all sufficiently large τ. In so doing, it is obvious that the right-hand side of (3.17) tends to infinity as $τ \to \infty$ . Thus, applying Lemma 3.3, we obtain $\begin{matrix} J_{R} (R, τ) \to 0 as τ \to \infty, \end{matrix}$ whence (3.22) follows an once. □

Proof of Theorem 2.1.

Let K be a compact subset of $R^{n}$ and $R > 0$ be a real number such that $K \subset B_{R / 2}$ . We denote $\begin{matrix} (3.26) & U (t) = \int_{B_{R / 2}} u (x, t) d x . \end{matrix}$ Since $u \in L_{1, loc} (R_{+}^{n + 1})$ , the right-hand side of (3.26) is defined for almost all $t \in (0, \infty)$ and, moreover, $U \in L_{1, loc} (0, \infty)$ . Let us put $\begin{matrix} U_{h} (τ) = \int_{0}^{\infty} ω_{h} (τ - t) U (t) d t, τ > h > 0, \end{matrix}$ where $ω_{h}$ is given by (3.1). We have $\begin{matrix} ‖ U_{h} - U ‖_{L_{1} (H)} \to 0 as h \to + 0 \end{matrix}$ for any compact set $H \subset (0, \infty)$ . Hence, there exists a sequence of positive real numbers ${h_{i}}_{i = 1}^{\infty}$ such that $h_{i} \to 0$ as $i \to \infty$ and $\begin{matrix} lim_{i \to \infty} U_{h_{i}} (τ) = U (τ) \end{matrix}$ for almost all $τ \in (0, \infty)$ . Also let $τ_{0} > R^{m}$ be a real number such that f is a non-negative function almost everywhere on $Q_{R}^{τ_{0} - R^{m}, \infty}$ . In view of (2.2), such a real number obviously exists.

Lemma 3.1 with $r_{1} = R / 2$ , $r_{2} = R$ , $τ_{1} = {(R / 2)}^{m}$ , and $τ_{2} = R^{m}$ yields $\begin{array}{l} \frac{1}{R^{m}} \int_{Q_{R}^{τ - R^{m}, τ + h_{i}}} u^{p} d x d t + \frac{1}{R^{m} - {(R / 2)}^{m}} \int_{Q_{R}^{τ - R^{m}, τ - {(R / 2)}^{m}}} u d x d t \\ (3.27) & ⩾ C \int_{Q_{R / 2}^{τ - h_{i}, τ + h_{i}}} ω_{h_{i}} (τ - t) u d x d t \end{array}$ for all $τ > τ_{0}$ and for all i such that $h_{i} < {(R / 2)}^{m}$ . Note that the first summand in the right-hand side of (3.2) is non-negative; therefore, it can be dropped.

Since $\begin{matrix} \int_{Q_{R / 2}^{τ - h_{i}, τ + h_{i}}} ω_{h_{i}} (τ - t) u d x d t = U_{h_{i}} (τ), τ > h_{i}, \end{matrix}$ passing to the limit as $i \to \infty$ in (3.27), we obtain $\begin{matrix} (3.28) & \frac{1}{R^{m}} \int_{Q_{R}^{τ - R^{m}, τ}} u^{p} d x d t + \frac{1}{R^{m} - {(R / 2)}^{m}} \int_{Q_{R}^{τ - R^{m}, τ - {(R / 2)}^{m}}} u d x d t ⩾ C U (τ) \end{matrix}$ for almost all $τ \in (τ_{0}, \infty)$ . By the Hölder inequality, $\begin{matrix} \int_{Q_{R}^{τ - R^{m}, τ - {(R / 2)}^{m}}} u d x d t ⩽ \int_{Q_{R}^{τ - R^{m}, τ}} u d x d t ⩽ C R^{(m + n) (p - 1) / p} {(\int_{Q_{R}^{τ - R^{m}, τ}} u^{p} d x d t)}^{1 / p} . \end{matrix}$ Thus, (3.28) implies the estimate $\begin{matrix} R^{- m} \int_{Q_{R}^{τ - R^{m}, τ}} u^{p} d x d t + R^{n - (m + n) / p} {(\int_{Q_{R}^{τ - R^{m}, τ}} u^{p} d x d t)}^{1 / p} ⩾ C U (τ) \end{matrix}$ for almost all $τ \in (τ_{0}, \infty)$ , whence in accordance with Lemma 3.5 we have $\begin{matrix} lim_{t \to \infty} \underset{τ \in (t, \infty)}{ess sup} U (τ) = 0 . \end{matrix}$

Theorem 2.1 is completely proved. □

Proof of Theorem 2.2.

We show that for any $R > 0$ there exists $τ > 0$ for which $u = 0$ almost everywhere on $Q_{R}^{τ, \infty}$ . Indeed, assume the converse. Let there be a real number $R > 0$ such that $\begin{matrix} J_{R} (R, τ_{i}) > 0, i = 1, 2, \dots, \end{matrix}$ for some unbounded sequence of positive real numbers ${τ_{i}}_{i = 1}^{\infty}$ . Using Lemma 3.3, we obtain $\begin{array}{l} \int_{J_{R} (R, τ_{i})}^{\infty} G^{- 1 / m} (ζ / 2) ζ^{1 / m - 1} d ζ + \int_{J_{R} (R, τ_{i})}^{\infty} \frac{d ζ}{G (ζ / 2)} + \int_{J_{R} (R, τ_{i})}^{\infty} \frac{d ζ}{G^{p} (ζ / 2)} \\ (3.29) & ⩾ C min {R \underset{Q_{2 R}^{τ_{i} - 2^{m} R^{m}, τ_{i}}}{ess inf} f^{1 / m}, R^{m} \underset{Q_{2 R}^{τ_{i} - 2^{m} R^{m}, τ_{i}}}{ess inf} f, R^{m p} \underset{Q_{2 R}^{τ_{i} - 2^{m} R^{m}, τ_{i}}}{ess inf} f^{p}} \end{array}$ for all sufficiently large i. At the same time, taking into account inequalities (3.23), (3.24), and (3.25) and the fact that G is a non-decreasing function on the interval $[0, \infty)$ with $G (0) > 0$ , we can assert that $\begin{array}{l} \int_{0}^{\infty} G^{- 1 / m} (ζ) ζ^{1 / m - 1} d ζ < \infty, \\ \int_{0}^{\infty} \frac{d ζ}{G (ζ)} < \infty, \end{array}$ and $\begin{matrix} \int_{0}^{\infty} \frac{d ζ}{G^{p} (ζ)} < \infty . \end{matrix}$ Thus, the left-hand side of (3.29) is bounded above by a constant that does not depend on i while the right-hand side tends to infinity as $i \to \infty$ . We arrive at a contradiction.

To complete the proof, it remains to note that a weak solution of inequality (1.1) can not vanish on a non-empty open subset of $R_{+}^{n + 1}$ on which the right-hand side of (1.1) is a positive function. □

Footnotes

Acknowledgements

The work of the second author is supported by RUDN University, Project 5-100.

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