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On a doubly degenerate parabolic equation with a nonlinear damping term
Boundary Value Problems volume 2021, Article number: 17 (2021)
Abstract
Consider a double degenerate parabolic equation arising from the electrorheological fluids theory and many other diffusion problems. Let \(v_{\varepsilon }\) be the viscous solution of the equation. By showing that \(|\nabla v_{\varepsilon }|\in L^{\infty }(0,T; L_{\mathrm{loc}}^{p(x)}(\Omega ))\) and \(\nabla v_{\varepsilon }\rightarrow \nabla v\) almost everywhere, the existence of weak solutions is proved by the viscous solution method. By imposing some restriction on the nonlinear damping terms, the stability of weak solutions is established. The innovation lies in that the homogeneous boundary value condition is substituted by the condition \(a(x)| _{x\in \partial \Omega }=0\), where \(a(x)\) is the diffusion coefficient. The difficulties come from the nonlinearity of \(\vert {\nabla v} \vert ^{p(x)-2}\) as well as the nonlinearity of \(|v|^{\alpha (x)}\).
1 Introduction
The initial-boundary value problem of a doubly degenerate parabolic equation
is considered, where \(p(x)>1\), \(\alpha (x)\) and \(a(x)\) are nonnegative \(C(\overline{\Omega })\) functions, \(f(x,t,v,|\nabla v|)\) is a continuous function and is called the nonlinear damping term. This equation comes from non-Newtonian fluid, the so-called electrorheological fluids, the heat conduction, and many other diffusion problems.
What first caught our attention is the heat conduction equation with a damping term
The author of [5] showed that the uniqueness is not true. The author of [31] and [33] generalized the results of [5] to a more general equation
where \(q\geq 1\), \(g(x)\geq 0\) and there is a point \(x_{0}\in \Omega \) such that \(g(x_{0})>0\). Based on these facts, one may conjecture that the heat conduction equation with a nonlinear damping term
is ill-posed.
The second aspect that attracted our attention is the so-called electrorheological fluids equation
which has been widely studied by many mathematicians, one can refer to [8, 9, 11–16] and the references therein. A more complicated equation
was studied in [2, 3]. Though the existence of weak solutions to equation (1.8) has been shown, the uniqueness result only for the case of \(|a(x,t,u)-a(x,t,v)|\leq \omega (|u-v|)\),
has been proved, where \(1<\beta <\frac{p^{+}}{p^{+}-1}\). In other words, the general uniqueness problem of equation (1.8) remains open till today.
Let \(a(x)\) satisfy
Then equation (1.1) is degenerate on the boundary ∂Ω. If \(\alpha (x)=0\), \(p(x)=p\) is a constant and \(f(x,t,v,|\nabla v|)=0\), on the stability of weak solutions, that the degeneracy of \(a(x)| _{x\in \partial \Omega }\) may take place of the usual boundary value condition (1.3) was revealed in [20, 21]. Moreover, whether
or
similar results have been obtained in [27] and [25] respectively. For the other related papers, one can refer to [19, 23, 24] etc.
For equation (1.1), compared with equation (1.7), there exists another diffusion coefficient \(a(x)\). Compared with equation (1.11), the convective term \(\sum_{i=1}^{N}\frac{\partial b_{i}(v)}{\partial x_{i}}\) is replaced by a nonlinear damping term \(f(x,t,v,|\nabla v|)\). Considering all these factors, compared the damping term \(f(x,t,v,|\nabla v|)\) with the degeneracy of \(a(x)|_{x\in \partial \Omega }\), the latter plays a leading role when the uniqueness problem is considered. Maybe such a conclusion can be explained by the fact that equation (1.1) represents the model that the diffusion process is more dominant than the damping phenomena. For example, for an epidemic model of diseases, it is impossible to know in advance that \(v=0\) on the boundary ∂Ω. Thus, imposing the boundary value condition (1.3) seems unreasonable, while the condition \(a(x)|_{ x\in \partial \Omega }=0\) can be explained as some anthropogenic interferences are made to control the epidemic across the border ∂Ω. In accord with this fact, in theory, we conjecture that under the condition \(a(x)|_{ x\in \partial \Omega }=0\), one can deduce that \(v=0\) on the boundary ∂Ω. This conjecture was partially proved in [22] several years ago, and we are not ready to discuss this conjecture in this paper for the time being.
The main aim of this paper is to establish the well-posedness theory for equation (1.1). To accomplish this aim, the nonlinearity of \(|v|^{\alpha (x)}\) and the nonlinearity of the damping term \(f(x,t,v,|\nabla v|)\) are the main difficulties to overcome. The extinction, the positivity, the large time behavior of the solutions and \(v=0\) on the boundary ∂Ω, all these important contents remain to be studied in the future.
Let us give the definition of weak solution.
Definition 1.1
If \(v(x,t)\) satisfies
and for any function \(\varphi \in C_{0}^{1}({Q_{T}})\),
and
for any \(\phi (x)\in C_{0}^{\infty }(\Omega )\), then we say that \(v(x,t)\) is a weak solution of equation (1.1) with initial value (1.2).
Here, the basic Banach space \(\mathbf{W}(Q_{T})\) and its dual space \(\mathbf{W}'(Q_{T})\) are defined by Antontsev and Shmarev in [2]. In addition, let
and set \(q(x)=\frac{p(x)}{p(x)-1}\) as usual. The main results in this paper are the following theorems.
Theorem 1.2
If \(a(x)\in C(\overline{\Omega })\) satisfies (1.10), \(f(x,t,v,|\nabla v|)\leq 0\) when \(v<0\),
and
-
(i)
when \(p^{-}\geq 2\),
$$ \bigl\vert f\bigl(x,t,v, \vert \nabla v \vert \bigr) \bigr\vert \leq c\bigl(a(x) \vert v \vert \bigr)^{\frac{2\alpha (x)}{p(x)}} \vert \nabla v \vert ^{2}, $$(1.16) -
(ii)
when \(p^{-}>1\),
$$ \bigl\vert f\bigl(x,t,v, \vert \nabla v \vert \bigr) \bigr\vert \leq c\bigl(a(x) \vert v \vert \bigr)^{\frac{\alpha (x)}{p(x)}} \vert \nabla v \vert . $$(1.17)
Then equation (1.1) with initial value (1.2) has a nonnegative solution \(v(x,t)\).
Theorem 1.3
Let \(u(x,t)\) and \(v(x,t)\) be two solutions of equation (1.1) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\) respectively and with the same homogeneous boundary value condition
If \(\alpha (x)\in C_{0}^{1}(\Omega )\), \(a(x)\in C(\overline{\Omega })\) satisfy (1.10) and the nonlinear damping term satisfies
-
(i)
when \(p^{-}\geq 2\),
$$ \begin{aligned} & \bigl\vert f\bigl(x,t,u, \vert \nabla u \vert \bigr)-f\bigl(x,t,v, \vert \nabla v \vert \bigr) \bigr\vert \\ &\quad \leq c \vert u-v \vert \bigl[\bigl(a(x) \vert v \vert \bigr)^{\frac{2\alpha (x)}{p(x)}} \vert \nabla v \vert ^{2}+\bigl(a(x) \vert u \vert \bigr)^{ \frac{2\alpha (x)}{p(x)}} \vert \nabla u \vert ^{2} \bigr], \end{aligned} $$(1.19) -
(ii)
when \(p^{-}>1\),
$$ \begin{aligned} & \bigl\vert f\bigl(x,t,u, \vert \nabla u \vert \bigr)-f\bigl(x,t,v, \vert \nabla v \vert \bigr) \bigr\vert \\ &\quad \leq c \vert u-v \vert \bigl[\bigl(a(x) \vert v \vert \bigr)^{\frac{\alpha (x)}{p(x)}} \vert \nabla v \vert +\bigl(a(x) \vert u \vert \bigr)^{ \frac{\alpha (x)}{p(x)}} \vert \nabla u \vert \bigr]. \end{aligned} $$(1.20)
Then
In particular, if \(\alpha (x)\equiv 0\), besides Theorem 1.3, we have the following theorem.
Theorem 1.4
Let \(u(x,t)\) and \(v(x,t)\) be two solutions of equation (1.1) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\) respectively and with the same homogeneous boundary value condition (1.18). If \(\alpha (x)\equiv 0\), \(p^{-}\geq 2\), \(a(x)\in C(\overline{\Omega })\) satisfy(1.10) and the nonlinear damping term satisfies
then
Moreover, since the diffusion coefficient \(a(x)\) satisfies (1.10), we can obtain a stability theorem without the boundary value condition (1.18).
Theorem 1.5
Let \(u(x,t)\) and \(v(x,t)\) be two solutions of equation (1.2) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\) respectively. If \(\alpha (x)\in C_{0}^{1}(\Omega )\), \(a(x)\) satisfies
and the nonlinear damping term satisfies (1.16) and (1.19), then the stability of weak solutions is true in the sense of (1.21).
Here and in what follows, \(\lambda >0\) is a small enough constant, and we define \(\Omega _{\lambda }=\{x\in \Omega : a(x)>\lambda \}\).
Compared with Theorem 1.3, there is not boundary value condition (1.18) in Theorem 1.4. Instead, condition (1.24) is imposed. Comparing with other related works [2, 3], the most distinctive assumption in this paper is that \(\alpha (x)\in C_{0}^{1}(\Omega )\). Since
is always true, and in particular \(\inf_{x\in \overline{\Omega }}{\alpha (x)}=0\), but \(\max_{x\in \overline{\Omega }}{\alpha (x)}\) can be larger than \(\frac{p^{+}}{p^{+}-1}\). This fact implies that when \(\alpha (x)\in C_{0}^{1}(\Omega )\), \(u^{\alpha (x)}\) is beyond the restriction (1.9). So, Theorem 1.3 and Theorem 1.4 have some essential improvements from the works [2, 3]. In the next research, we will try to do some work when \(\alpha (x)\) is not limited to \(C_{0}^{1}(\Omega )\). By the way, from [4, 5] [33] and [31], in order to obtain the well-posedness of weak solutions to equation (1.1), the damping term \(f(x,t,u,\nabla u)\) must satisfy some restrictions, for example, condition (1.19) and condition (1.20) in our paper. A similar condition was first introduced by Karlsen and Ohlberger in their paper [10], in which the uniqueness of weak solutions to the equation
is proved. Although, as one of the reviewers pointed out, condition (1.19) or condition (1.20) is reasonable, are there other conditions to replace condition (1.19) or condition (1.20)? This is also an interesting problem.
2 The proof of Theorem 1.2
In this section, we prove Theorem 1.2.
Let us consider an approximate problem
where \(v_{\varepsilon 0} \in C^{\infty }_{0}(\Omega )\), \(|v_{\varepsilon 0}|_{L^{\infty }(\Omega )}\leq |v_{0}|_{L^{\infty }( \Omega )}\), \(a(x) \vert \nabla v_{\varepsilon 0} \vert ^{(x)p}\) converges to \(a(x)|\nabla v_{0}(x)|^{p}\) in \({L^{1}}(\Omega )\) uniformly. Since \(f(x,t,v_{\varepsilon },|\nabla v_{\varepsilon }|)\leq 0\) when \(v_{\varepsilon }<0\) and satisfies (1.16) or (1.17), the above problem (2.1)–(2.3) has a unique nonnegative solution \(v_{\varepsilon }\in L^{\infty }(0,T; W^{1,p}_{\mathrm{loc}}(\Omega ))\), and
one can refer to [3, 7, 18]for details.
Lemma 2.1
If \(u_{\varepsilon }\in L^{\infty }(0,T;L^{2}(\Omega ))\cap \mathbf{W}(Q_{T})\), \(\| u_{\varepsilon t}\|_{\mathbf{W}'(Q_{T})}\leq c\), \(\|\nabla (|u_{\varepsilon }|^{q-1}u_{\varepsilon })\|_{p,Q_{T}}\leq c\), then there is a subsequence of \(\{u_{\varepsilon }\}\) which is relatively compact in \(L^{s}(Q_{T})\) with \(s\in (1,\infty )\). Here, \(q\geq 1\).
This lemma can be found in [17].
Proof of Theorem 1.2
At first, let us multiply (2.1) by \(v_{\varepsilon }\). Since \(f(x,t,v,|\nabla v|)\leq 0\) when \(v<0\) and satisfies (1.16), by the Young inequality, we have:
-
(i)
when \(p^{-}\geq 2\), \(f(x,t,v_{\varepsilon },|\nabla v_{\varepsilon }|)\) satisfies (1.16), we have
$$ \begin{aligned} &\frac{1}{2} \int _{\Omega }v_{\varepsilon }^{2}\,dx+ \iint _{{Q_{T}}} \bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{\varepsilon } \vert ^{\alpha (x)}+\varepsilon \bigr) \vert \nabla v_{\varepsilon } \vert ^{p(x)}\,dx\,dt \\ &\quad =\frac{1}{2} \int _{\Omega }v_{\varepsilon 0}^{2}\,dx+ \iint _{{Q_{T}}}f(x,t,v_{ \varepsilon },|\nabla v_{\varepsilon })|v_{\varepsilon }\,dx\,dt \\ &\quad \leq \frac{1}{2} \int _{\Omega }v_{\varepsilon 0}^{2}\,dx+c \iint _{{Q_{T}}}\bigl(a(x) \vert v \vert \bigr)^{ \frac{2\alpha (x)}{p(x)}} \vert \nabla v \vert ^{2}\,dx\,dt \\ &\quad \leq \frac{1}{2} \int _{\Omega }v_{\varepsilon 0}^{2}\,dx+ \iint _{{Q_{T}}} \bigl[\varepsilon a(x) \vert v_{\varepsilon } \vert ^{\alpha (x)} \vert \nabla v_{ \varepsilon } \vert ^{p(x)}+c(\varepsilon ) \bigr]\,dx\,dt \\ &\quad \leqslant c; \end{aligned} $$(2.5) -
(ii)
when \(p^{-}>1\), \(f(x,t,v_{\varepsilon },|\nabla v_{\varepsilon }|)\) satisfies (1.17), we have
$$ \begin{aligned} &\frac{1}{2} \int _{\Omega }v_{\varepsilon }^{2}\,dx+ \iint _{{Q_{T}}} \bigl(a(x)+\varepsilon \bigr) \bigl( \vert v_{\varepsilon } \vert ^{\alpha (x)}+\varepsilon \bigr) \vert \nabla v_{\varepsilon } \vert ^{p(x)}\,dx\,dt \\ &\quad =\frac{1}{2} \int _{\Omega }v_{\varepsilon 0}^{2}\,dx+ \iint _{{Q_{T}}}f(x,t,v_{ \varepsilon },|\nabla v_{\varepsilon }) \biggl\vert v_{\varepsilon }\,dx\,dt \\ &\quad \leq \frac{1}{2} \int _{\Omega }v_{\varepsilon 0}^{2}\,dx+c \iint _{{Q_{T}}}\bigl(a(x) \vert v \vert \bigr)^{ \frac{\alpha (x)}{p(x)}} \biggr\vert \nabla v|\,dx\,dt \\ &\quad \leq \frac{1}{2} \int _{\Omega }v_{\varepsilon 0}^{2}\,dx+ \iint _{{Q_{T}}} \bigl[\varepsilon a(x) \vert v_{\varepsilon } \vert ^{\alpha (x)} \vert \nabla v_{ \varepsilon } \vert ^{p(x)}+c(\varepsilon ) \bigr]\,dx\,dt \\ &\quad \leqslant c. \end{aligned} $$(2.6)
Then
which implies
Secondly, according to the definition of Banach space \(\mathbf{W}(Q_{T})\) [2], \(C_{0}^{\infty }(Q_{T})\) is dense in \(\mathbf{W}(Q_{T})\). Now, for any \(u\in C_{0}^{\infty }(Q_{T})\), \(\|u\|_{W(Q_{T})}=1\), we have
According to condition (1.16) or (1.17), by the Young inequality, we easily deduce
and
Let \(\lambda >0\) be a small enough constant, set \(D_{\lambda }=\{x\in \Omega : \text{dist}(x,\partial \Omega )>\lambda \}\), and let \(\varphi \in C_{0}^{\infty }(\Omega )\), \(0\leq \varphi \leq 1\) satisfy
Then
and
as well as
or equivalently,
for any \(s\in (1,\infty )\). By (2.13)–(2.14), \(\varphi v_{\varepsilon }^{\frac{\alpha (x)}{p(x)}+1}\) is relatively compact in \(L^{s}(Q_{T})\). Then \(\varphi v_{\varepsilon }^{\frac{\alpha (x)}{p(x)}+1}\rightarrow \varphi v_{1}\) a.e. in \(Q_{T}\). Due to the arbitrariness of λ, we know \(v_{\varepsilon }^{\frac{\alpha (x)}{p(x)}+1}\rightarrow v_{1}\) a.e. in \(Q_{T}\).
By (2.4), \(v \in L^{\infty }(Q_{T})\), and
it must be
Thus, \(v_{\varepsilon }\rightarrow v\) a.e. in \(Q_{T}\).
Moreover, since \(a(x)\) is positive in Ω, (2.8) yields
Now, we want to show the local integral of ∇v. For any \(\phi (x)\in C_{0}^{1}(\Omega )\), if we choose \((v_{\varepsilon }^{\frac{\alpha (x)}{p(x)}+1}-v^{ \frac{\alpha (x)}{p(x)}+1} )\phi \) as the test function, then
We have the following facts:
and
which goes to zero as \(\varepsilon \rightarrow 0\). By (2.17)–(2.18), we can deduce that
which implies that
and we have
Since \(|v_{\varepsilon }|^{\alpha (x)} \vert \nabla v_{\varepsilon } \vert ^{p(x)-2} \nabla v_{\varepsilon }\in L^{1}(0,T; L_{\mathrm{loc}}^{\frac{p(x)}{p(x)-1}}( \Omega ))\), we can deduce the local integral of ∇v, i.e.,
For any large enough n, m, \(v_{n}=v_{\varepsilon }|_{\varepsilon =\frac{1}{n}}\) and \(v_{m}=v_{\varepsilon }|_{\varepsilon =\frac{1}{m}}\) are two viscous solutions. Then
Egoroff’s theorem yields, for fixed \(\delta >0\), a closed set \(E_{\delta }\subset Q_{T}\) such that the measure \(\mu (Q_{T}-E_{\delta })\leq \delta \) and \(v_{n}\rightrightarrows v\) uniformly on \(E_{\delta }\). By drawing the methods of [26–28], we can extrapolate that
from which we can deduce \(\nabla v_{n}\rightarrow \nabla v\) a.e. in \(Q_{T}\). Thus, we have
In the end, the initial value is true in the sense of (1.14) can be shown as that of [1]. Thus, v is a weak solution of equation (1.1) in the sense of Definition 1.1. □
3 The global stability
For small \(\eta >0\), we define \(g_{\eta }(x)\) to be an odd function, when \(s\geq 0\), \(g_{\eta }(x)\) has the form
Proceeding as in [28], we can prove the following lemma, we omit the details here.
Lemma 3.1
Let \(u\in \mathbf{W}(Q_{T})\), \(u_{t}\in \mathbf{W}'(Q_{T})\). Then \(\forall \textit{ a.e. } t_{1}, t_{2}\in (0, T)\),
The following lemma is the basic characteristics of the variable exponent Sobolev spaces [6, 12, 32].
Lemma 3.2
-
(i)
The spaces \((L^{p(x)}(\Omega ), \|\cdot \|_{L^{p(x)}(\Omega )} )\), \((W^{1,p(x)}(\Omega ), \|\cdot \|_{W^{1,p(x)}(\Omega )} )\), and \(W^{1,p(x)}_{0}(\Omega )\) are reflexive Banach spaces.
-
(ii)
The \(p(x)\)-Hölder inequality. Let \(p(x)\) and \(q(x)\) be real functions with \(\frac{1}{p(x)}+\frac{1}{q(x)} = 1\). Then, for any \(u \in L^{p(x)}(\Omega )\) and \(v \in L^{q(x)}(\Omega )\), we have
$$ \biggl\vert \int _{\Omega }uv \,dx \biggr\vert \leq 2 \Vert u \Vert _{L^{p(x)}(\Omega )} \Vert v \Vert _{L^{q(x)}(\Omega )}. $$ -
(iii)
\(\|u\|_{L^{p(x)}(\Omega )} \) and \(\int _{\Omega }|u|^{p(x)} \,dx\) satisfy
$$ \begin{aligned} & \textit{If } \Vert u \Vert _{L^{p(x)}(\Omega )} = 1,\quad {\textit{then }} \int _{ \Omega } \vert u \vert ^{p(x)} \,dx = 1. \\ &\textit{If } \Vert u \Vert _{L^{p(x)}(\Omega )} > 1,\quad {\textit{then }} \Vert u \Vert ^{p^{-}}_{L^{p(x)}( \Omega )}\leq \int _{\Omega } \vert u \vert ^{p(x)} \,dx\leq \Vert u \Vert ^{p^{+}}_{L^{p(x)}( \Omega )}. \\ &\textit{If } \Vert u \Vert _{L^{p(x)}(\Omega )} < 1, \quad {\textit{then }} \Vert u \Vert ^{p^{+}}_{L^{p(x)}( \Omega )}\leq \int _{\Omega } \vert u \vert ^{p(x)} \,dx\leq \Vert u \Vert ^{p^{-}}_{L^{p(x)}( \Omega )}. \end{aligned} $$
Theorem 3.3
Let \(u(x,t)\) and \(v(x,t)\) be two solutions of equation (1.1) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\) respectively and with the same homogeneous boundary value condition (1.18). If \(\alpha (x)\in C_{0}^{1}(\Omega )\), the nonlinear damping term satisfies
-
(A)
\(p^{-}\geq 2\),
$$ \begin{aligned} &\bigl\vert f\bigl(x,t,u, \vert \nabla u \vert \bigr)-f \bigl(x,t,u, \vert \nabla v \vert \bigr) \bigr\vert \\ &\quad \leq c f_{1}(x,t) \vert u-v \vert \bigl[ \vert v \vert ^{\frac{2\alpha (x)}{p(x)}} \vert \nabla v \vert ^{2}+ \vert u \vert ^{ \frac{2\alpha (x)}{p(x)}} \vert \nabla u \vert ^{2} \bigr], \end{aligned} $$(3.1) -
(B)
\(p^{-}>1\),
$$ \begin{aligned} &\bigl\vert f\bigl(x,t,u, \vert \nabla u \vert \bigr)-f \bigl(x,t,u, \vert \nabla v \vert \bigr) \bigr\vert \\ &\quad \leq c f_{1}(x,t) \vert u-v \vert \bigl[ \vert v \vert ^{\frac{\alpha (x)}{p(x)}} \vert \nabla v \vert + \vert u \vert ^{ \frac{2\alpha (x)}{p(x)}} \vert \nabla u \vert \bigr], \end{aligned} $$(3.2)and one of the following conditions is true:
-
(i)
$$ a(x)^{-1}f_{1}(x,t)\leq c, $$(3.3)
-
(ii)
there is a constant \(r\geq 2+\frac{2}{p(x)-2}\) such that
$$ \iint _{Q_{T}}a(x)^{1-r}f_{1}(x,t)^{r}\,dx\,dt \leq c, $$(3.4)then
$$ \int _{\Omega } \bigl\vert u(x,t) - v(x,t) \bigr\vert \,dx \leq c \int _{\Omega } \bigl\vert u_{0}(x) - v_{0}(x) \bigr\vert \,dx, \quad \textit{a.e. } t\in [0,T). $$
-
(i)
Proof
We only give the proof of case (A). Case (B) can be proved in a similar way, we omit the details.
Since \(u(x,t)\) and \(v(,t)\) satisfy the same homogeneous boundary value condition (1.18), we can choose \(g_{\eta }(u - v)\) as the test function. Then
There are two facts much in evidence in (3.5). One is that, by Lemma 3.1, we have
Another one is that, by the monotonicity of the operator \(|\nabla u|^{r-2}\nabla u\), we have
Let us discuss the other terms in (3.5). In the first place, \(\alpha (x)\in C_{0}^{1}(\Omega )\), we set \(\Omega _{\alpha }=\{x\in \Omega :\alpha (x)>0\}\) and define
Since (2.23) yields \(|\nabla u|^{p(x)}, |\nabla v|^{p(x)}\in L_{\mathrm{loc}}^{1}(Q_{T})\), using the fact \(\lim_{\eta \rightarrow 0}g'_{\eta }(s)s=0\) and the Lebesgue dominated convergence theorem, we have
and similarly
According to (3.8)–(3.9), we can obtain
where \(p^{+}\) and \(q^{+}\) follow from (iii) of Lemma 3.2.
In the second place, since the nonlinear damping term satisfies (3.1), using the Hölder inequality, we have:
(i) By (3.3),
where \(p_{21}=\max_{x\in \overline{\Omega }}\frac{p(x)}{2}\) or \(\min_{x\in \overline{\Omega }}\frac{p(x)}{2}\) according to (iii) of Lemma 3.2, \(p_{22}\) has a similar sense.
(ii) Since \(r\geq 2+\frac{2}{p(x)-2}\), there is \(\frac{p(x)}{p(x)-2}\frac{(r-1)p(x)-2r}{p(x)}\geq 1\). By (3.4), there are two constants \(l_{1}>1\), \(l_{2}>1\) such that
Now, let \(\eta \rightarrow 0\) in (3.5). According to (3.11)–(3.12), there is a constant \(l_{3}>1\) such that
By a generalized Gronwall inequality [26], we have the conclusion. □
Proof of Theorem 1.3
If the nonlinear damping term satisfies (1.19)
we easily show that there is a constant \(l>1\) such that
Proceeding as in the proof of Theorem 3.3, we have the conclusion. If the nonlinear damping term satisfies (1.20), we can prove the conclusion in a similar way, and we do not repeat the details here. □
Theorem 3.4
Let \(p^{-}\geq 2\), \(u(x,t)\) and \(v(x,t)\) be two solutions of equation (1.1) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\) respectively and with the same homogeneous boundary value condition (1.18). If \(\alpha (x)\equiv 0\), the nonlinear damping term satisfies
and condition (3.3) or (3.4) is true, then
Proof
Since \(u(x,t)\) and \(v(,t)\) satisfy the same homogeneous boundary value condition (1.18), we can choose \((u - v)\) as the test function. Then
By Lemma 3.1, we have
and
At the same time, since the nonlinear damping term satisfies (3.14), using the Hölder inequality, we have the following:
(i) By (3.3), there is a constant \(l>1\) such that
where \(p_{21}=\max_{x\in \overline{\Omega }}\frac{p(x)}{2}\) or \(\min_{x\in \overline{\Omega }}\frac{p(x)}{2}\) according to (iii) of Lemma 3.2, \(p_{22}\) has a similar sense.
(ii) Since \(r\geq 2+\frac{2}{p(x)-2}\), there is \(\frac{p(x)}{p(x)-2}\frac{(r-1)p(x)-2r}{p(x)}\geq 1\). By (3.14), there are constants \(l_{1}>1\), \(l_{2}>1\), and \(l_{3}>1\) such that
According to (3.18)–(3.19), there is a constant \(l_{4}>1\) such that
By a generalized Gronwall inequality [26], we have the conclusion. □
Proof of Theorem 1.4
Since the nonlinear damping term satisfies
proceeding as in the proof of Theorem 3.4, we have the conclusion. □
4 The global stability if \(\int _{\Omega }a(x)^{1-p(x)}\,dx<\infty \)
Recalling that, by a weak characteristic function \(\chi (x)\) of Ω, \(\chi (x)\in C(\overline{\Omega })\) and
we can set another weak characteristic function as
In this section, we explore the stability of weak solutions by the weak characteristic function method [29, 30].
Theorem 4.1
Suppose that \(u(x,t)\) and \(v(x,t)\) are two solutions of equation (1.2) with the initial values \(u_{0}(x)\), \(v_{0}(x)\) respectively. If there is a weak characteristic function \(\chi (x)\in C^{1}(\overline{\Omega })\) satisfying
\(\alpha (x)\in C_{0}^{1}(\Omega )\), the nonlinear damping term satisfies (3.1) and one of (3.3) (3.4), then
Proof
Since \(\alpha (x)\in C_{0}^{1}(\Omega )\), as before we set \(\Omega _{\alpha }=\{x\in \Omega :\alpha (x)>0\}\) and
as well as
By choosing \(g_{\eta }(u - v)\phi _{\lambda }(x)\) as a test function, since
we have
As usual, we now analyze every term in (4.4). In the first place, we have
and
In the second place, since \(|\nabla u|^{p(x)}, |\nabla v|^{p(x)}\in L_{\mathrm{loc}}^{1}(Q_{T})\), using the fact \(\lim_{\eta \rightarrow 0}g'_{\eta }(s)s=0\) and the Lebesgue dominated convergence theorem, we have
thus
Here, \(p_{1}\) is \(p^{+}\) or \(p^{-}\) according to (iii) of Lemma 3.2, \(q_{1}\) is \(q^{+}\) or \(q^{-}\).
In the third place, we denote \(D_{\lambda }=\{x\in \Omega : \chi (x)>\lambda \}\). If we choose λ small enough, then for \(\Omega _{\alpha }=\{x\in \Omega : \alpha (x)>0\}\),
there is a constant \(c_{\alpha }\) such that \(\phi (x)>c_{\alpha }\) provided \(x\in \overline{\Omega }_{\alpha }\). According to the definition of the weak characteristic function \(\chi (x)\),
If we define that
then
and
In the fourth place, since
by (4.3),
as \(\lambda \rightarrow 0\). Here, we have used the fact
At last, for the nonlinear damping term satisfying (3.1), we can deal with it as in Theorem 3.3, we omit the details here.
Letting \(\eta \rightarrow 0\) in (4.4), let \(\lambda \rightarrow 0\). The Gronwall inequality yields the conclusion. □
Proof of Theorem 1.5
Only if we choose \(\chi (x)=a(x)\), then
and the conclusion follows clearly. □
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Xu, W. On a doubly degenerate parabolic equation with a nonlinear damping term. Bound Value Probl 2021, 17 (2021). https://doi.org/10.1186/s13661-021-01493-x
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DOI: https://doi.org/10.1186/s13661-021-01493-x