Initial boundary value problem for a class of $p$-Laplacian equations with logarithmic nonlinearity

1.   Introduction

In this paper, we consider the following $p$-Laplacian equations with logarithmic nonlinearity.

where $1 < p < 2$, $1 < q\leq2$, $r > 1$, $\beta, \lambda > 0$, $T\in(0, +\infty]$, $\Omega\in\mathbb{R}^N$ is a bounded domain with smooth boundary and $u_0(x)\in L^\infty(\Omega)\cap W^{1, p}_0(\Omega)$ is a nonzero non-negative function.

Problem (1.1) is a class of parabolic equation with logarithmic nonlinearity, it is worth pointing out that the interest in studying problem (1.1) relies not only on mathematical purposes, but also on their significance in real models. Among the fields of mathematical physics, biosciences and engineering, problem (1.1) is one of the most important nonlinear evolution equations. For example, in the combustion theory, we can use the function $u(x, t)$ to represent temperature, the $-{\rm{div}}(|\nabla u|^{p-2}\nabla u)$ term to represent thermal diffusion, $\beta|u|^{q-2}u$ to represent absorption, and $\lambda |u|^{r-2}u\ln{|u|}$ to be the source. In the diffusion theory, we can use $u(x, t)$ to represent the density of a type of population at position $x$ at time $t$, the $-{\rm{div}}(|\nabla u|^{p-2}\nabla u)$ term represents the diffusion of density, $\lambda |u|^{r-2}u\ln{|u|}$ and $\beta|u|^{q-2}u$ represents the absorption and the sources, respectively. We refer the reader to ^[1,2] and the references therein for further details on more practical applications of problem (1.1).

The research with logarithmic nonlinearity terms is the current research hotspot. The literature on the evolution equations with logarithmic nonlinearity term is very interesting, we refer the readers to ^[3,4,5,6] and the references therein. At the same time, the study of $p$-Laplacian equations has also achieved many important results. The study of $p$-Laplacian equations can be divided into two cases, namely $1 < p < 2$ and $p > 2$. For the case of $p > 2$, most researchers discuss the global existence and blow-up of solutions of the equations (see ^[7,8,9,10]). For the case of $1 < p < 2$, the extinction and attenuation estimation of solutions are mainly discussed, we refer the readers to ^[11,12,13].

In particular, there are also some papers concerning properties such as global existence or extinction for the problem (1.1) for special cases.

In ^[14], Liu studied a more general form

where $1 < p < 2$, $q\leq1$, $r > 1$, $\beta, \lambda > 0$, $\Omega\subset\mathbb{R}^N$ ($N > 2$) is a bounded domain with smooth boundary and $u_0(x)\in L^\infty(\Omega)\cap W^{1, p}_0(\Omega)$ is a nonzero non-negative function. The author gave the extinction properties and attenuation estimates of the solutions by comparison principle and differential inequality.

In ^[15], Cao and Liu considered the following initial-boundary value problem for a nonlinear evolution equation with logarithmic source

where $1 < p < 2$, $u_0\in H^1_0(\Omega)$, $T\in(0, +\infty)$, $k\geq0$, $\Omega\subset\mathbb{R}^n(n\geq1)$ is a bounded domain with smooth boundary $\partial\Omega$. They proved the global existence of weak solutions and studied the asymptotic behavior of solutions and gave some decay estimates and growth estimates by constructing a family of potential wells and using the logarithmic Sobolev inequality.

In ^[16], Pan et al. considered the following pseudo-parabolic equation with $p$-Laplacian and logarithmic nonlinearity terms

in a bounded domain $\Omega\subset\mathbb{R}^n(n\geq1)$ with smooth boundary, where $u_0\in W^{1, p}_0(\Omega)\backslash\{0\}$ and the parameters $p, q$ satisfy $2 < p < q < p(1+\frac{2}{n})$. They gave the upper and lower bound estimates of blow-up time and blow-up rate, and established a weak solution with high initial energy. In ^[17], Xiang and Yang studied the following initial-boundary value problem for the fractional $p$-Laplacian Kirchhoff type equations

where $0 < s < 1 < p < 2$, $1 < q\leq2$, $r > 1$, $\mu, \lambda > 0$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $M: [0, \infty)\rightarrow (0, \infty)$ is a continuous function. By flexible application of differential inequalities, they gave the extinction and the decay estimates of solutions.

Inspired by the above work, we study problem (1.1). Compared with problem (1.2) and (1.5), the focus of our work is partial differential equations with logarithmic nonlinearity. If the nonlinear term $\lambda |u|^{r-2}u\ln{|u|}$ in problem (1.1) is transformed into $\lambda |u|^{r-2}u$, then the problem (1.1) can be transformed into problem (1.2). For more research results of the logarithmic nonlinear $p$-Laplacian equations, we refer the readers to ^{[18,19,20,21]} and the references therein. But as far as we know, no work has dealt with the global existence and extinction properties of solutions for problem (1.1) with both the absorption and source effects as well as the term of logarithmic nonlinearity. To state our main results, we need the following two definitions.

Definition $1.1$ (Weak solution). A function $u(x, t)$ is said to be a weak solution of problem (1.1), if $(x, t)\in\Omega\times[0, T)$, $u\in L^p\big(0, T; W^{1, p}_0(\Omega))\cap C(0, T; L^\infty(\Omega)\big)$, $u_t\in L^2\big(0, T; L^2(\Omega)\big)$, $u(x, 0) = u_0(x)\in W^{1, p}_0(\Omega)$, for all $v\in W^{1, p}_0(\Omega)$, $t\in(0, T)$, the following equation holds

where $(\cdot, \cdot)$ means the inner product of $L^2(\Omega)$.

Definition $1.2$ (Extinction of solutions). Let $u(t)$ be a weak solution of problem (1.1). We call $u(t)$ is an extinction of solutions if there exists $T > 0$ such that $u(x, t) > 0$ for all $t\in(t, T)$ and $u(x, t)\equiv0$ for all $t\in[T, +\infty)$.

Further, we respectively define the energy functional $J(u)$ and the Nehari functional $I(u)$ of problem (1.1) as

and

By the subsequent Lemma 2.1 and a simple calculation, we can obtain

for $0 < \sigma < 2^* -r$. So $J(u)$ and $I(u)$ are well-defined for $u\in W^{1, p}_0(\Omega)$.

Next, the potential well $W$ and its corresponding set $V$ are defined by

Let

and define the Nehari manifold

Moreover, we define

Next, we state our main results.

Theorem $1.1$ (Global existence). Assume that $u_0(x)\in L^\infty(\Omega)\cap W^{1, p}_0(\Omega)$. Then problem (1.1) admits a global weak solution $u\in L^p\big(0, T; W^{1, p}_0(\Omega))\cap C(0, T; L^\infty(\Omega)\big)$ with $u_t\in L^2\big(0, T; L^2(\Omega)\big)$ for $0\leq t < +\infty$.

Theorem $1.2$ (Globally bounded and blow-up). Let $u(x, t)$ be the weak solution of problem (1.1) and $r = p$. If $J(u_0)\leq d$, $I(u_0)\geq 0$ and $p = q$, then the weak solution $u(x, t)$ is globally bounded. Moreover, if $J(u_0) < 0$ and $p > q$, the weak solution $u(x, t)$ blows up at $+\infty$.

Theorem $1.3$ Assume that $I(u_0) > 0$, $r = p$ and $q = 2.$

(1) If $\lambda < R_0$, then the weak solution of $(1.1)$ satisfies

(2) If $2N/(N+2s) < p < 2$ and $\lambda < R_0$ or $1 < p\leq 2N/(N+2s)$ and $\lambda < R_1$, then the nonnegative solutions of $(1.1)$ vanish in finite time, and

for $2N/(N+2) < p < 2$, and

for $1 < p < 2N/(N+2)$, where

and $C$ is the embedding constant, $\Gamma(p, \Omega)$ will be given in section 3.

Theorem $1.4$ Let $r = p$ and $p > q$. If $0 < J(u_0) < R_2$ and $I(u_0) < 0$, then the solution $u(x, t)$ of $(1.1)$ is non-extinct in finite time, where

Theorem $1.5$ Assume $I(u_0) > 0$, $r > p$ and $q = 2$, then the nonnegative weak solution of problem $(1.1)$ vanishes in finite time and

for $2N/(N+2)\leq p < 2$, $\lambda < R_2$, and

for $1 < p < 2N/(N+2s)$, $\lambda < R_3 = \frac{E_0}{E_1}$, where

and $C_{p^*}$ is the embedding constant, $l_1, v_2, s_2$ will be given in section 3.

The paper is organized as follows. In section 2, we give some necessary Lemmas such as some properties for Nehari functional and known results for ODEs. In section 3, we present the proof of the main theorems.

2.   Preliminaries and Lemmas

Lemma $2.1$ (^[22]) Let $\alpha$ be positive number, then

Lemma $2.2$ Let $u\in W^{1, p}_0(\Omega)\backslash \{0\}$, then

where $\Gamma(p, \Omega): = \frac{|\Omega|}{ep}+\frac{1}{e(p^*-p)}C^{p^*}_{p^*}$, $C_{p^*}$ is the best constant of embedding from $W^{1, p}_0(\Omega)$ to $L^{p^*}(\Omega).$

Proof. For convenience, we provide complete proof. As we know, for $1 < p < 2$,

Let $\Omega_1 = \{x\in\Omega:|u(x)|\leq\|u\|_{1, \Omega} \}$ and $\Omega_2 = \{x\in\Omega:|u(x)| > \|u\|_{1, \Omega} \}$, we can obtain

Using the properties of logarithmic, we have

Taking $\sigma = p^*-p$ in Lemma 2.1, and by Sobobev embedding inequality, we obtain

By (2.2), (2.3) and (2.4), we get

By direct calculation and Eq (2.1), we have

The proof is completed.

Lemma $2.3$ (^[23]) Let $p > 1$, $\mu > 0$, and $u\in W^{1, p}_0(\Omega)$, then we have

where

Lemma $2.4$ Let $u\in W^{1, p}_0\backslash\{0\}$ and $r = p$, then we have

(ⅰ) If $0 < \|\nabla u\|_p\leq R$, then $I(u)\geq0$;

(ⅱ) If $I(u) < 0$, then $\|\nabla u\|^p > R$;

where $R = \lambda^{\frac{1}{p}}_1\Big(\frac{p^2e}{\lambda n\mathbb{L}_p}\Big)^{\frac{n}{p^2}}$ and $\lambda_1$ is the first eigenvalue of the following equation

Proof. (ⅰ) By (1.3) and Lemma 2.3, we can obtain

Taking $\mu = \frac{1}{\lambda}p$ in (2.5), we get

If $0 < \|\nabla u\|_p\leq R$, then $\|u\|_p\leq\lambda^{-\frac{1}{p}}_1\|\nabla u\|_p\leq\Big(\frac{p^2e}{\lambda n\mathbb{L}_p}\Big)^{\frac{n}{p^2}}$, we have $I(u)\geq0$.

(ⅱ) If $I(u) < 0$, by (2.6), we have

namely

The proof is completed.

Lemma $2.5$ (^[24]) If $1\leq p_0 < p_\theta < p_1\leq\infty$, then we have

for all $u\in L^{p_0}(\Omega)\cap L^{p_1}(\Omega)$ with $\theta\in(0, 1)$ defined by $\frac{1}{p_\theta} = \frac{1-\theta}{p_0}+\frac{\theta}{p_1}$.

Lemma $2.6$ (^[25]) Let $y(t)$ be a non-negative absolutely continuous function on $[T_0, +\infty)$ satisfying

where $\alpha, \beta > 0$ are constants and $k\in(0, 1)$. Then

where $T_* = \frac{1}{(1-k)\beta}\ln\big(1+\frac{\beta}{\alpha}y^{1-k}(T_0)\big).$

Lemma $2.7$ Assume that $J(u_0)\leq d$, then the sets $\mathcal{N_+}$ and $\mathcal{N_-}$ are both invariant for $u(t)$, i.e, if $u_0\in \mathcal{N_-}$ (resp. $u_0\in\mathcal{N_+}$), then $u(t)\in \mathcal{N_-}$ (resp. $u(t)\in\mathcal{N_+}$) for all $t\in[0, T).$

Proof. We only prove the case of $u(t)\in \mathcal{N_-}$, and the proof of $u(t)\in \mathcal{N_+}$ is similar.

Step 1: $J(u_0) < d$. If $u(t)\not\in \mathcal{N_-}$, there exists a $t_0 > 0$, such that

By (2.1), we have

where $S$ is the embedding constant, $0 < \sigma\leq p^*-r$. Note that $I(u)\leq0$, we have $\|u\|_{1, \Omega} = \|\nabla u\|_p > \big(\frac{\sigma}{\lambda S^{r+\sigma}_{r+\sigma}}\big)^{\frac{1}{r+\sigma-p}} > 0,$ which implies $\|u(t_0)\|_{1, \Omega} = \|\nabla u(t_0)\|_p > \big(\frac{\sigma}{\lambda S^{r+\sigma}_{r+\sigma}}\big)^{\frac{1}{r+\sigma-p}} > 0$, so we get $u(t_0)\in \mathcal{N}$. Choosing $\nu = u_t$ in Definition 1.1 and integrating with respect to time $0$ to $t$, we can obtain

namely

While by the definition of $d$ in (1.10), we get $J(u(t_0))\geq d$, which gives a contradiction.

Step 2: $J(u_0) = d$. Similarly, we assume that the conclusion is incorrect, then it exists a $t_1 > 0$, such that $I(u(t_1)) = 0, \ {\rm{and}} \ I(u(t)) < 0\ {\rm{for\; all}} \ t\in[0, t_1).$ By calculation of (2.7), we get $\|\nabla u\|_p > 0$, which implies $u(t_1)\in \mathcal{N}$. Since $\frac{1}{2}\frac{d}{dt}\|u\|^2_2 = -I(u(t))$ for all $t\in[0, t_1)$, combining with boundary conditions, we obtain $u_t\neq0$. By (2.8), we have

which gives a contradiction with the definition of $d$.

3.   Proof of main results

In this section, we prove that the main results of problem (1.1).

Proof of Theorem 1.1

First we let $\{\omega_j(x)\}$ be the basis function of $W^{1, p}_0(\Omega)$. Next, we construct the following approximate solutions $u_{m}(t)$ of problem (1.1) as follows:

which satisfy

and

where $j = 1, 2, ..., m$, and $\xi_{jm} = (u_{m}(0), \omega_j)$ are given constants. We use $(\cdot, \cdot)$ to represent the inner product in $L^{2}(\Omega)$. The standard theory of ODEs, e.g. Peano's theorem, yields that $g_{jm}(t)\in C^1([0, \infty); W^{1, p}_0(\Omega))$ and $g_{jm}(0) = \xi_{jm}$, thus $u_m\in C^1([0, \infty); W^{1, p}_0(\Omega))$.

Next, we try to get a priori estimates of the approximate solutions $u_m$. Multiplying (3.1) by $g'_{jm}(t)$, summing for $j$ from 1 to $m$ and integrating with respect to time from 0 to $t$, we can obtain

For sufficiently large $m$ and by (3.3), we have

where

Multiplying (3.1) by $g_{jm}(t)$, then summing $j$ from 0 to $m$, we obtain

By Lemma 2.1, we have

For $\sigma\in[1, 2-p)$, namely

which implies

By (3.3), (3.4) and (3.6), we can obtain

where

Therefore, by (3.6) and (3.7), there is a function $u\in L^p\big(0, T; W^{1, p}_0(\Omega))\big)$ with $u_t\in L^2\big(0, T; L^2(\Omega)\big)$, and a subsequence of $\{u_m\}^\infty_{m = 1}$ (still denoted by $\{u_m\}^\infty_{m = 1}$) such that for $t\in(0, \infty)$, as $m\rightarrow \infty$, we obtain

Since $W^{1, p}_0\hookrightarrow L^p(\Omega)$ is compact, by (3.8), (3.9) and using Aubin-Lions compactness theorem, we have

which implies $u_m\rightarrow u \ {\rm{a.e.\ in}}\ \Omega\times(0, \infty)$, and then $|u_m|^{p-2}u_m\ln|u_m|\rightarrow |u|^{p-2}u\ln|u|$ a.e. in $\Omega\times(0, \infty).$

Fixing $j$ in (3.1) and letting $m\rightarrow \infty$, we get

which implies

for all $v\in W^{1, p}_0(\Omega)$. The next work is to prove that $\chi(t) = |\nabla u|^{p-2}\nabla u$, that is to say, we should change Eq (3.12) into the following equation

The remainder of the proof is the same as that in ^[15].

Proof of Theorem 1.2

We first consider the case $0 < J(u_0) < d$ and $I(u_0) > 0$. Choosing $\nu = u$ in Definition 1.1, we obtain

which implies

Taking $\nu = u_t$ in Definition 1.1 and integrating with respect to time $0$ to $t$, we get

Next, we proof that $u(x, t)\in W$ for any $t > 0$. If there exists a $t_0 > 0$, such that $u(x, t_0)\in \partial W$, namely

By (3.15), we get $J(u(x, t_0)) = d$ is not true. If $u(x, t_0)\in\mathcal{N}$, then by the definition of $d$ in (1.10), we get $J(u(x, t_0))\geq d$, which also contradict with (3.15). So we have $u(x, t)\in W$.

By (1.5), (1.7) and (3.15), we have

Note that $I(u) > 0$, we obtain

Through (3.14) and a simple calculation, we have the following inequality

By Lemma 2.4, we have

Thus, combining with the above inequality, we know that the weak solution of problem (1.1) is globally bounded.

Now we consider $J(u_0) = d$. Take a function $\rho_m$ which satisfies $\rho_m > 0$ and $\lim\limits_{n\rightarrow +\infty}\rho_m = 1$. Let $u(x, 0) = u_{0m}(x) = \rho_mu_0(x), x\in\Omega$, for the following equations

Since $u_0(x)\in L^\infty(\Omega)\cap W^{1, p}_0(\Omega)$ is a nonzero non-negative function, $\rho_m\in(0, 1)$ and $I(u_0)\geq0$, then we have

and

Note that $r = p$ and $p = q$, we have

If $\int_\Omega|u_0|\ln|u|dx < 0$, by (3.22), we can obtain

If $\int_\Omega|u_0|\ln|u|dx\geq0$, by (3.20) and (3.22), we can obtain

Moreover, by a simple calculation, we have

thus, we get

which implies that $J(\rho_mu_0)$ is strictly increasing with respect to $\rho_m$. Then we have

From the results above, we can derive that the weak solution of Eq (3.19) is globally bounded.

Then, we discuss that weak solution blows up at infinity. Let $M(t) = \frac{1}{2}\int_\Omega|u(x, t)|^2dx$, then we have

By $(3.26)$ and the following equation

which implies

Making $\nu = u_t$ in Definition 1.1, we get

By a simple calculation, we get

Thus, we can obtain

By (3.27) and (3.28), we have

namely

Therefore, the following inequality holds

The proof is completed.

Proof of Theorem 1.3

(1) Choosing $\nu = u$ in Definition 1.1, we get

By Lemma 2.2 and (3.29), we have

Combining (3.30) and Lemma 2.4, we can obtain

Note that $\lambda < R_0$, it follows that

so we have

Therefore, we conclude that $\|u(\cdot, t)\|_2\rightarrow0 \ \rm{as} \ t\rightarrow \infty.$

(2) We consider first the case $2N/(N+2) < p < 2$ with $\lambda < R_0.$ Multiplying (1.1) by $u$ and integrating over $\Omega$, we have

By the first eigenvalue $\lambda_1$ and Lemma 2.4, we obtain

By virtue of Holder's inequality and the embedding theorem, we obtain

where $C$ is the embedding constant.

By (3.32) and (3.33), we get the following differential inequality

where

Setting $y(t) = \|u(\cdot, t)\|^2_2$, $y(0) = \|u_0(\cdot)\|^2_2$, by Lemma 2.2, we obtain

where

We now turn to the case $1 < p\leq2N/(N+2)$ and $\lambda < R_1.$

Multiplying (1.1) by $u^l\ (l = \frac{2N-(1+N)p}{p}\geq1)$ and integrating over $\Omega$, we can obtain

By the embedding theorem and the specific choice of $l$, we have

where $C$ is the embedding constant. Thus (3.37) becomes

where

Setting $y(t) = \|u(\cdot, t)\|_{l+1}$, $y(0) = \|u_0(\cdot)\|_{l+1}$, by Lemma 2.6, we can obtain

where

The proof is completed.

Proof of Theorem 1.4

We first define

then, we obtain

and

By (3.28) and (3.41), we can obtain

Not that

Choosing $\mu = \frac{1}{\lambda}p$ in (3.43), we have

Since $I(u_0) < 0$, we get

By (3.42), (3.44) and $p > q$, we obtain

where $R_2 = \lambda\frac{1}{p^2}\big(\frac{p^2e}{\lambda n\mathbb{L}_p}\big)^{\frac{n}{p}}$. Multiplying both sides by $M(t)$ in inequality (3.45), we get

Since

thus we have

namely

So, there exists a finite time $T_0 > 0$ such that

which implies

The proof is completed.

Proof of Theorem 1.5

We consider first the case $p < r < 2$ and $2N/(N+2) < p < 2$. Choosing $\nu = u$ in Definition 1.1, we have

By Lemma 2.2, we have

Since $I(u_0) > 0$, combining Lemma 2.4 and Lemma 2.7, we get

To facilitate discussion, we let

Note that $s_1 > r$ and $v_1\in(0, 1)$, by the Hölder's inequality, the Sobolev embedding theorem and Lemma 2.5, we have

Further, by the Young inequality for any $\varepsilon > 0$, we get

By (3.49) and (3.50), we can obtain

namely

where

Taking $\varepsilon = \Big(\frac{\beta}{3\lambda\ln(R)|\Omega|C^{r(1-v_1)}_{p^*}}\Big)^{\frac{p-r(1-v_1)}{r(v_1-1)}}$ in $(3.51)$, we have

where

Since $\lambda < R_2$, thus $C_4 > 0.$ By Lemma 2.6, we can obtain

where

When $1 < p\leq2N/(N+2)$, $p < r\leq2$ and $\lambda < R_3$. Taking $\nu = u^l\ (l = \frac{2N-(1+N)p}{p}\geq1)$ in Definition 1.1, we obtain

By Lemma 2.1, we have

where $p_1 = l+p+\sigma-1$. Let

Thus we have $l_0 > l_1 > p$, $v_1\in(0, 1)$ and

By the Hölder's inequality, the following inequality holds

Further, by the Young inequality for any $\varepsilon > 0$, we get

Substituting (3.57) into (3.56), we get

where

Taking $\varepsilon = \frac{1}{3}\frac{lp^pe\sigma}{(p+l-1)^p}|\Omega|^{\frac{l_1-s_2}{s_2}}C^{v_2-1}_{p^*}$ in (3.58), we can obtain

where

Since $l_0 < p^*$, by the Sobolev embedding theorem and the Hölder's inequality, we get

Combining (3.59) with (3.60), we obtain

where

Since $\lambda < R_3$, thus $C_7 > 0.$ By Lemma 2.6, we can obtain

where

The proof is completed.

4.   Conclusions

In this work, we study the initial boundary value problem for a class of $p$-Laplacian diffusion equations with logarithmic nonlinearity. Compared with the research in literature ^[26], we further discussed the integer-order Laplacian equations when $1 < p < 2$, and proved the global existence of the solution of problem (1.1) by the Galerkin approximation method. Compared with problems (1.2) and (1.5), we give the extinction and attenuation estimates of the weak solution of problem (1.1) by using potential well theory and Nehari manifold. In addition, we also prove that the weak solution of problem (1.1) is globally bounded and blows up at infinity. In the next work, we will further discuss the properties of the solution of Eq (1.1) when $r\neq p$ and $q\neq 2$, and study the diffusion $p(x)$-Laplacian with logarithmic nonlinearity.

Acknowledgments

This research was supported by the project of Guizhou province science and technology plan under (No.Qiankehe foundation-ZK [2021] YIBAN317), and by the project of Guizhou Minzu University under (No.GZMU [2019] YB04).

Conflict of interest

The authors declare no conflict of interest.