Decay estimate and non-extinction of solutions of p-Laplacian nonlocal heat equations

1.   Introduction

The study of differential equations and variational problems with nonstandard $p(x)-$growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see ^{[1,2,3,4,5,6,7,8,9]}).

In this paper, we consider the Neumann problem to the following initial parabolic equation with logarithmic source:

where $\Omega$ is a bounded domain in $\mathbb{R} ^{N}$ with smooth boundary $\partial \Omega,$ $p$ $\in$ $\left(2, +\infty \right)$, $\oint_{\Omega }u_{0}dx = \frac{1}{\left\vert \Omega \right\vert } \int_{\Omega }u_{0}dx = 0$ with $u_{0}\neq 0.$

Problem (1.1) has been studied by many other authors in a more general form

where $\Omega$ is a bounded domain in $\mathbb{R} ^{N}$ $(N\geq 1)$ with $\left\vert \Omega \right\vert$ denoting its Lebesgue measure, $N$ is the outer normal vector of $\partial \Omega,$ and the function $f(u)$ is usually taken to be a power of $u.$

Wang. M, Wang, .Y in ^[10], studied the properties of positive solutions when $f(u) = \left\vert u\right\vert ^{p}.$ The authors showed global existence and exponential decay in the case where $\left\vert \Omega \right\vert$ $\leq k$ and they obtained a blow-up result under the assumption that the initial data is bigger than some "Gaussian function" in the case where $\left\vert \Omega \right\vert$ $> k$.$.$ When $f(u) = u\left\vert u\right\vert ^{p}$ and $\int_{\Omega }udx > 0,$ non-global existence result is discussed by ^[11].

C. Qu, X. Bai, S. Zheng ^[12] considered the nonlocal p-Laplace equation

where a critical blow-up solution is determined by $q$ and the sign of the initial energy.

More recently, L. Yan, Z. Yong ^[13] established a blow-up and non-extinction of solutions under appropriate conditions for (1.1) in the case $p = 2$.

Apart the aforesaid attention given to polynomial nonlinear terms, logarithmic nonlinearity has also received a great deal of interest from both physicists and mathematicians (see for example ^{[14,15,16,17,18]}). This type of nonlinearity was introduced in the nonrelativistic wave equations describing spinning particles moving in an external electromagnetic field and also in the relativistic wave equation for spinless particles ^[19]. Moreover, the logarithmic nonlinearity appears in several branches of physics such as inflationary cosmology ^[20], nuclear physics ^[21], optics ^[22] and geophysics ^[23]. With all those specific underlying meaning in physics, the global-in-time well-posedness of solution to the problem of evolution equation with such logarithmic type nonlinearity captures lots of attention. Birula and Mycielski (^[24,25]) studied the following problem:

which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit $p$ goes to $1$ for the $p$ -adic string equation (^[26]). In ^[27], Cazenave and Haraux considered

and established the existence and uniqueness of the solution for the Cauchy problem. Gorka ^[28] used some compactness arguments and obtained the global existence of weak solutions, for all

to the initial-boundary value problem (1.4) in the one-dimensional case. Bartkowski and Gorka, ^[29] proved the existence of classical solutions and investigated the weak solutions for the corresponding one-dimensional Cauchy problem for Equation (1.5). Hiramatsu et al. ^[30] introduced the following equation

to study the dynamics of Q-ball in theoretical physics and presented a numerical study. However, there was no theoretical analysis for the problem. In ^[31], Han proved the global existence of weak solutions, for all

to the initial boundary value problem (1.6) in $\mathbb{R}^{3}.$

Motivated by the above studies, in this paper we investigate a blow up, non existence and decay of solutions of problem (1.1).

It is necessary to note that the presence of the logarithmic nonlinearity causes some difficulties in deploying the potential well method. In order to handle this situation we need the following logarithmic Sobolev inequality which was introduced in ^[10].

Lemma 1. Let $p > 1, \mu > 0,$ and $u\in W^{1, p}\left(\mathbb{R} ^{n}\right) \diagdown \left\{ 0\right\}.$ Then we have

where

2.   Preliminaries

2.1.   Notations and some inequalities

We begin this section by introducing some notations that will be used throughout the paper

for $1 < p < +\infty.$We also we define $X_{0} = W_{0}^{1, p}(\Omega)\diagdown \left\{ 0\right\}.$

Lemma 2. Let $\varrho$ be a positive number. Then the following inequality holds

Lemma 3. (a) For any function $u\in W_{0}^{1, p}(\Omega)$, we have the inequality

for all $q\in \left[ 1, \infty \right)$ if $n\leq p,$ and $1\leq q\leq \frac{np}{n-p}$ if $n > p.$ Then the best constant depends $B_{q, p}$ only on $\Omega, n, p$ and $q.$

We will denote the constant $B_{p, p}$ by $B_{p.}$

(b) Let $2\leq p < q < p^{\ast }$. For any $u\in W_{0}^{1, p}(\Omega)$ we have

where $C$ is a positive constant and

Remark 1. It follows from Lemma 2 that

Now we considering the functional $J$ and $I$ defined on $X_{0}$ as follows

The functions $I$ and $J$ are continuous (they are defined as in ^[32] with some modifications). Moreover, we have

Then it is obvious that

$\mathcal{\phi } = \left\{ u\in X_{0}:I(u) = 0, \text{ }\left\Vert u\right\Vert _{p}^{p}\neq 0\right\}.$

$d = \underset{u\in \phi }{\inf }J(u).$

$M = \frac{R^{p}}{p^{2}}.$

From ^[33], we know $d\geq M.$

$\mathcal{N}_{\delta } = \left\{ u\in X_{0}:I_{\delta }(u) = 0\right\}.$

Theorem 1. $\left(\mathit{\text{Local existence}}\right)$ Let $u_{0}\in X_{0}.$ Then there exists a positive constant $T_{0}$ such that the problem (1.1) has a weak solution $u(x, t)$ on $\Omega \times$ $(0, T_{0})$. Furthermore, $u(x, t)$ satisfies the energy inequality

Lemma 4. Suppose that $\theta > 0,$ $\alpha > 0,$ $\beta > 0$ and $h(t)$ is a nonnegative and absolutely continuous function satisfying $h^{^{\prime }}(t)+\alpha h^{\theta }(t)\geq \beta,$ then for $0 < t < \infty$, it holds

Lemma 5. If $0 < J(u_{0}) < E_{1} = \frac{1}{p^{2}}e^{\frac{b}{p}},$ where $b = n\log \left(\frac{p^{2}e}{n\mathcal{L}_{p}}\right),$, then there exists a positive constant $\alpha _{2} > \alpha _{1}$ such that

Proof. Using the logarithmic Sobolev inequality in Lemma 1 and $\mu = p$, we have

Denote $\alpha = \left\Vert u\right\Vert _{p},$ $b = n\log \left(\frac{p^{2}e}{ n\mathcal{L}_{p}}\right),$ we have

Let $h^{^{\prime }}(\alpha _{1}) = 0,$ $E_{1} =$ $h(\alpha _{1}) = \frac{1}{ p^{2}}e^{\frac{b}{p^{2}}}$

Furthermore, we get $h(\alpha)$ is increasing in $(0, \alpha _{1})$ and decreasing in $(\alpha _{1}, \infty)$. Since $J(u_{0}) < E_{1}$, there exists a positive constant $\alpha _{2} > \alpha _{1}$ such that $J(u_{0})$ = $h(\alpha _{2})$. Let $\alpha _{0}$ $= \left\Vert u_{0}\right\Vert _{2}$, from (2.6) and (2.7), we have

Since $\alpha _{0}$, $\alpha _{2}$ $\geq \alpha _{1}$, we get $\alpha _{0}\geq \alpha _{2}$, so (2.5) holds for $t = 0$.

To prove (2.5) for $t > 0$, we assume the contrary that $\left\Vert u(., t)\right\Vert _{2} <$ $\alpha _{2}$ for some $t_{0} > 0$. By the continuity of $\left\Vert u(., t)\right\Vert _{2}$ and $\alpha _{1} < \alpha _{2}$, we may choose $t_{0}$ such that $\left\Vert u(., t_{0})\right\Vert _{2}$ $>$ $\alpha _{1}$, then it follows from (2.6)

which contradicts the fact that $J(u)$ is nonincreasing in $t$ by (2.4), so (2.5) is true.

Lemma 6. Let $H(u) = E_{1}-J(u), J(u_{0}) < E_{1}$, then $H(u)$ satisfies the following estimates

Proof. It is obvious that $H(u)$is nondecreasing in $t$, by (2.4), then it follows from $J(u_{0}) < E_{1}$that

2.2.   Potential well

Let $u\in X_{0}$ and consider the real function $j$ $:\lambda \rightarrow$ $J(\lambda u)$ for $\lambda > 0$,

The following Lemma shows that $j(\lambda)$ has a unique positive critical point $\lambda ^{\ast } = \lambda ^{\ast }(u)$ see $\left[ 3\right].$

Lemma 7. Let $u\in X_{0}.$Then it holds

(1) $\underset{\lambda \rightarrow 0^{+}}{\lim j(\lambda)} = 0$ and $\underset {\lambda \rightarrow +\infty }{\lim j(\lambda)} = -\infty,$

(2) there is a unique $\lambda ^{\ast } = \lambda ^{\ast }(u) > 0$ such that $j^{^{\prime }}(\lambda ^{\ast }) = 0,$

(3) $j(\lambda)$ is increasing on $\left(0, \lambda ^{\ast }\right),$ decreasing on $\left(\lambda ^{\ast }, +\infty \right)$ and attains the maximum at $\lambda ^{\ast },$

(4) $I(\lambda u) > 0$ for $0 < \lambda < \lambda ^{\ast },$ $I(\lambda u) < 0$ for $\lambda ^{\ast } < \lambda < +\infty$ and $I(\lambda ^{\ast }u) = 0.$

Proof. For $u\in X_{0},$ by the definition of $j$, It is clear that (1) holds due to $\left\Vert u\right\Vert _{p}\neq 0,$ and by derivation of $j$, we have

which implies that

the statements of (2) and (3) can be shown easily. The last property, (4), is only a simple corollary of the fact that

The proof of lemma is complete.

Next we denote

Lemma 8. (1) if $I(u) > 0$ then $0 < \left\Vert u\right\Vert _{p} < R$,

(2) if $I(u) < 0$ then $\left\Vert u\right\Vert _{p} > R, \qquad$

(3) if $I(u) = 0$ then $\left\Vert u\right\Vert _{p}\geq R$.

Proof. By the definition of $I(u),$ we have

Choosing $\mu = p$, and we apply the logarithmic Sobolev inequality (Lemma 1), we obtain

if $I(u) > 0,$ then

that's mean

and if $I(u) < 0,$ we obtain

property (3) we can argue similarly the proof of (2).

The proof of lemma is complete.

3.   Global existence and decay estimates

Lemma 9. (see ^[34]) Let $f$ : $\mathbb{R} ^{+}\rightarrow \mathbb{R} ^{+}$ be a nonincreasing function and $\sigma$ is a nonnegative constant such that

Then we have

(a) $f(t)\leq f(0)e^{1-\omega t},$ for all $t\geq 0,$ whenever $\sigma = 0,$

(b) $f(t)\leq f(0)\left(\frac{1+\sigma }{1+\omega \sigma t}\right) ^{\frac{1 }{\sigma }},$ for all $t\geq 0,$ whenever $\sigma > 0.$

Remark 2. As in ^[33], we introduce the following set:

Theorem 2. if $u_{0}\in W_{1}^{+}, 0 < J(u_{0}) < M^{^{\prime }} = \frac{R}{p^{2}},$ Then the solution $u(x, t)$ of problem (1.1) admits a global weak solution such that

satisfying the energy estimate

Moreover, the solution decays polynomially, namely

where $C_{s}$ and $\zeta _{s}$ are positives constants $.$

Proof. Existence of global weak solutions

It suffices to show that $\left\Vert \nabla u\right\Vert _{p}^{p}$ and $\left\Vert u\right\Vert _{p}^{p}$ are bounded independent of $t$.

In the space $W_{0}^{1, p}(\Omega)$, we take a basis $\{w_{j}\}_{_{j = 1}}^{ \infty }$ and define the finite dimensional space

Let $u_{0m}$ be an element of $V_{m}$ such that

as $m\rightarrow +\infty,$ We find the approximate solution $u_{m}(x, t)$ of the problem (1.1) in the form

where the coefficients $\alpha _{mj}(1\leq j\leq m)$ where ($\alpha _{mj}(0) = a_{m, j}),$ satisfy the system of ordinary differential equations

We multiply both sides of (3.2) by $\alpha _{mi}^{^{\prime }}(t),$and we take the sum, we get

that's mean

this implies that

we deduce

by integrating (3.3) with respect to $t$ on $[0, t]$, we obtain the following equality

where $T_{m}$ is the maximal existence time of solution $u_{mt}(x, t).$

It follows from (3.1), (3.4), and the continuity of $J$ that

with $J(u_{0}) < d$ and

for m large sufficiently large $m,$ We will show that

and for sufficiently large $m$, and assume that (*) does not hold and let $t_{\ast }$ be the smallest time for which $u_{m}(t_{\ast })\notin W_{1}^{+}.$ Then, by the continuity of $u_{m}(t_{\ast })\in \partial W_{1}^{+}$, we have

Nevertheless, it is clear that (3.6)$_{1}$ could not occur by (3.5) while if (3.6)$_{2}$ holds then, by the definition of $d$, we have

which also contradicts with (3.5). Thus, (*) is true.

On the other hand, since $u_{m}(t)\in W_{1}^{+},$ and

we deduces from (3.5) that

for sufficiently large $m$ and $t\in$ $[0, T_{m})$. Further, by the logarithmic inequality, we have

This implies that

Taking $\mu < p,$ we deduce that

Decay estimates

We define

for $u(t)\in W_{1}^{+}$ by using (2.3) and the energy inequality that, we know

By using the logarithmic Sobolev inequality in Lemma 1 and we put $\mu = p,$ we have

where $l_{2, p}$ is a constant in the embedding $L^{p}(\Omega)\hookrightarrow L^{2}(\Omega), p > 2$ and $\zeta = \frac{1}{l_{2, p}}\log \frac{ M}{J(u_{0})}.$

From (3.8), we have

Combining (3.10) and (3.11), it follows that

Let $T\rightarrow +\infty$ and apply Lemma 9, such that $f(t) = \left\Vert u(t)\right\Vert _{2}^{2}, \sigma = \frac{p-2}{2}, f(0) = 1, \omega = \frac{1}{ 2\varsigma }$

we obtain the following decay estimate

where $C_{s}$ is positive constant and $\zeta _{s} = \frac{1}{4\varsigma }.$

4.   Blow up of weak solutions

4.1.   Blow up at $+\infty$

Definition 1. (Blow-up at$+\infty$) Let $u(x, t)$ be a weak solution of (1.1). We call $u(x, t)$ blow-up at $+\infty$ if the maximal existence time $T =$ $+\infty$ and

Theorem 3. Assume $J(u_{0}) < 0,$ then the solution $u(x, t)$ of problem (1.1) is blow-up at $+\infty$. Moreover, if $\left\Vert u_{0}\right\Vert _{2}\leq \left(\frac{-pJ(u_{0})}{l_{2, p}}\right) ^{p},$ the lower bound for blow-up rate can be estimated by

which is independent of $t$.

Proof. By the definition of $J(u)$ and (2.4), $M(t)$ satisfies

by using (2.4) defined in theorem 1 and the condition $J(u_{0}) < 0,$ we have

so by (*) and (**), we get

And by the definition of weak solution, we know that $u\in W^{1, p}(\Omega)$ For any $t_{0} > 0$, we claim that

Otherwise, there exists $t_{0} > 0$ such that $\int_{0}^{t_{0}}\left\Vert u_{s}\right\Vert _{2}^{2}ds = 0,$ and hence $u_{t} = 0$ for a.e., $(x, t)\in \Omega \times \left(0, t_{0}\right).$ Then it follows from (4.2) that

for a.e., $t\in (0, t_{0})$, and then we get from (2.1)

Combining it with $J(u)\leq J(u_{0})\leq 0$, we obtain$\left\Vert \ u\right\Vert _{p} = 0$ for all $t\in \lbrack 0, t_{0}]$, which contradicts the definition of $u$. Then (4.3) follows.

Fix $t_{0} > 0$ and let $\delta = \int_{0}^{t}\left\Vert \ u\right\Vert _{2}^{2}ds$, then we know that $\delta$ is a positive constant. Integrating (4.2) over $(t_{0}, t)$, we obtain

We have

where $H(t)$ defined in lemma 6

Hence

And from (4.2), we know

we have

where $l_{2, p}$ is a constant in the embedding $L^{p}\left(\Omega \right) \hookrightarrow L^{2}\left(\Omega \right),$ $p > 2.$

By using Lemma 2.1, $J(u_{0}) < 0$, and $\left\Vert u_{0}\right\Vert _{2}^{2}$ $\leq \left(\frac{-pJ(u0)}{l_{2, p}}\right) ^{\frac{^{2}}{p}}$, we have

which means (4.1) is true.

4.2.   Non-extinct in finite time

Definition 2. (Finite time blow-up) Let $u(x, t)$ be a weak solution of (1.1).We call $u(x, t)$ blow-up in finite time if the maximal existence time $T$ is finite and

Lemma 10. Let $\phi$ be a positive, twice differentiable function satisfying the following conditions

for some $\overline{t}\in \left[ 0, T\right),$ and the inequality

where $\alpha > 1.$ Then we have

with $\sigma$ is a positive constant, and

This implies

Theorem 4. Assume $0 < J(u_{0}) < M$ and $u\in W_{1}^{-}$, then the solution $u(x, t)$ of problem (1.1) is non-extinct in finite time, defined by

Proof. we define the functional

Then one has

and

by using (2.4) in theorem 1, we have

by lemma 8 for $I(u) < 0,$ which implies

by (4.13) and (4.14), we get

where $M = \frac{R}{p^{2}}.$

In other hand we have

also, we have

Now, multiplying (4.15) by $\Gamma (t),$ we get

by using (4.17) in (4.18), we obtain

This follows

By virtue of lemma 10, where $\alpha = \frac{p}{2} > 1,$ and $\phi (t) = \Gamma (t),$ we get

there exists $T_{\ast } > 0$ such that

which implies

therefore, we get

This ends the proof.

5.   Conclusions

In this work, by using the logarithmic Sobolev inequality and potential wells method, we study the initial boundary value problem of a nonlocal heat equations with logarithmic nonlinearity in a bounded domain, where we obtain the decay, blow-up and non-extinction of solutions under some conditions, and the results extend the results of a recent paper Lijun Yan and Zuodong Yang ^[13]. In our next study, we will try to apply an alternative approach using the variational principle that has been presented in previous studies ^[35].

Conflict of interest

The authors declare no conflict of interest.