The initial-boundary value problem of the evolutionary \(p(x)\)-Laplacian equation
$$ u_{t} =\operatorname{div}\bigl( \vert \nabla u \vert ^{p(x)-2}\nabla u\bigr),\quad (x,t) \in {Q_{T}} = \Omega \times (0,T), $$
(1.1)
has been widely studied [1–6]. It is well-known that the equation arises in many applications in the electrorheological fluids, physics and biology [1–3]. Here, \(\Omega\subset \mathbb{R}^{N}\) is a bounded domain with smooth boundary ∂Ω, \(p(x)\in C^{1}(\overline{\Omega})\), and we denote \(p^{+}=\max_{x\in \Omega}p(x), p^{-}=\min_{x\in \Omega}p(x)>1\). Let \(d(x)=\operatorname{dist}(x,\partial \Omega)\) be the distance function, the constant \(\alpha>0\). The well-posedness of the solutions of the equation
$$ u_{t} =\operatorname{div}\bigl(d^{\alpha}(x) \vert \nabla u \vert ^{p-2}\nabla u\bigr),\quad (x,t) \in {Q_{T}} , $$
(1.2)
was first studied by Yin-Wang [7], and later by Zhan-Xie [8] et al. A similar equation related to the \(p(x)\)-Laplacian
$$ u_{t} =\operatorname{div}\bigl(d^{\alpha}(x) \vert \nabla u \vert ^{p(x)-2}\nabla u\bigr),\quad (x,t) \in {Q_{T}}, $$
(1.3)
was studied by Zhan-Wen [9, 10] recently. In [9], the stability of the weak solutions is proved in a similar way as that of [7]. But there remained a gap when \(p^{-}-1\leq \alpha\leq p^{+}-1\). In [10], if \(p(x)\) is required to satisfy the logarithmic Hölder continuity condition
$$\bigl\vert p(x)-p(y) \bigr\vert \leq \omega\bigl( \vert x-y \vert \bigr), \quad \forall x,y\in \Omega, \vert x-y \vert < \frac{1}{2}, $$
with
$$\mathop {\overline {\lim }}\limits _{s \to {0^{+} }} \omega (s)\ln \biggl(\frac{1}{s}\biggr) = C < \infty, $$
by complicate calculations, the gap had been filled up.
In this paper, we will establish the well-posedness of the solutions of equation
$$ u_{t} =\operatorname{div}\bigl(d^{\alpha}(x) \vert \nabla u \vert ^{p(x)-2}\nabla u\bigr)+\frac{\partial b_{i}(u,x,t)}{\partial x_{i}},\quad (x,t) \in {Q_{T}}, $$
(1.4)
with the initial value
$$ u|_{t=0} = u_{0}(x),\quad x\in\Omega, $$
(1.5)
but without any boundary value condition. The initial-boundary value problem of equation (1.4) was first considered by the author in [11], it was shown that the convection term \(\frac{\partial b_{i}(u,x,t)}{\partial x_{i}}\) may influence the boundary value condition. We conjectured that, to ensure the well-posedness of the solutions, a partial boundary value condition should be imposed on equation (1.4). From then on, I had spent much time to consider the problem, and found that it is difficult to determine which part of the boundary should be imposed the boundary value. Thereupon, in this paper, we turn our attention to a study of the well-posedness of the solutions without any boundary value condition. We will introduce a new kind of the weak solutions matching up with equation (1.4), and try to prove the uniqueness of the new weak solutions only dependent on the initial value.
We denote
$$ W^{1,p(x)}_{\alpha}=\biggl\{ u\in W^{1, p(x)}_{\mathrm{loc}}( \Omega): \int_{\Omega}d^{\alpha}(x) \vert \nabla u \vert ^{p(x)}\,dx< \infty\biggr\} . $$
(1.6)
Clearly,
$$ W^{1,p(x)}_{\alpha}\subseteq W^{1, p(x)}_{\mathrm{loc}}( \Omega). $$
(1.7)
Here \(W^{1, p(x)}(\Omega)\) is the variable exponent Sobolev space, one can refer to [12–14] for the details. Some basic properties of the space are quoted in the following lemma.
Lemma 1.1
-
(i)
The spaces
\((L^{p(x)}(\Omega), \Vert \cdot \Vert _{L^{p(x)}(\Omega)} )\), \((W^{1,p(x)}(\Omega), \Vert \cdot \Vert _{W^{1,p(x)}(\Omega)} )\)
and
\(W^{1,p(x)}_{0}(\Omega)\)
are reflexive Banach spaces.
-
(ii)
\(p(x)\)-Hölder’s inequality. Let
\(q_{1}(x)\)
and
\(q_{2}(x)\)
be real functions with
\(\frac{1}{q_{1}(x)}+\frac{1}{q_{2}(x)} = 1\)
and
\(q_{1}(x) > 1\). Then the conjugate space of
\(L^{q_{1}(x)}(\Omega)\)
is
\(L^{q_{2}(x)}(\Omega)\). And for any
\(u \in L^{q_{1}(x)}(\Omega)\)
and
\(v \in L^{q_{2}(x)}(\Omega)\), we have
$$\biggl\vert \int_{\Omega}uv \,dx \biggr\vert \leq 2 \Vert u \Vert _{L^{q_{1}(x)}(\Omega)} \Vert v \Vert _{L^{q_{2}(x)}(\Omega)}. $$
-
(iii)
$$ \begin{aligned} & \textit{If } \Vert u \Vert _{L^{p(x)}(\Omega)} = 1,\quad \textit{then } \int_{\Omega} \Vert u \Vert ^{p(x)} \,dx = 1, \\ & \textit{if } \Vert u \Vert _{L^{p(x)}(\Omega)} > 1, \quad \textit{then } \Vert u \Vert ^{p^{-}}_{L^{p(x)}(\Omega)}\leq \int_{\Omega} \vert u \vert ^{p(x)} \,dx\leq \Vert u \Vert ^{p^{+}}_{L^{p(x)}(\Omega)}, \\ & \textit{if } \Vert u \Vert _{L^{p(x)}(\Omega)} < 1, \quad \textit{then } \Vert u \Vert ^{p^{+}}_{L^{p(x)}(\Omega)}\leq \int_{\Omega} \vert u \vert ^{p(x)} \,dx\leq \Vert u \Vert ^{p^{-}}_{L^{p(x)}(\Omega)}. \end{aligned} $$
The new kind of the weak solutions matching up with equation (1.4) is defined as follows.
Definition 1.2
A function \(u(x,t)\) is said to be a solution of equation (1.4) with the initial condition (1.5), if
$$ u \in L^{\infty}(Q_{T}),\qquad u_{t}\in L^{2}(Q_{T}),\qquad d^{\alpha}{ \vert {\nabla u} \vert ^{p(x)}} \in {L^{\infty}}\bigl(0,T;L^{1}(\Omega) \bigr), $$
(1.8)
and
$$ \iint_{Q_{T}} \biggl[u_{t}(\varphi_{1} \varphi_{2}) + d^{\alpha}(x) \vert \nabla u \vert ^{p(x)- 2}\nabla u \cdot \nabla (\varphi_{1}\varphi_{2}) +b_{i}(u,x,t)\frac{\partial (\varphi_{1}\varphi_{2})}{\partial x_{i}}\biggr] \,dx\,dt = 0, $$
(1.9)
where \(\varphi_{2} \in C_{0}^{1} (Q_{T})\) as usual, but \(\varphi_{1}\) only satisfies, for any given t, \(\varphi_{1}(x,t)\in W^{1,p(x)}_{\alpha}\), and, for any given x, \(\vert \varphi_{1}(x,t) \vert \leq c\). The initial condition (1.5) is satisfied in the sense of that
$$ \lim_{t\rightarrow 0} \int_{\Omega}\bigl\vert u(x,t) - u_{0}(x) \bigr\vert \,dx = 0. $$
(1.10)
A basic result of the existence of the solution is the following.
Theorem 1.3
If
\(p^{-}>2\)
and
\(0<\alpha<\frac{p^{-}-2}{2}\), \(\beta>0\), \(b_{i}(s,x,t)\)
and its partial derivatives satisfy the condition
$$ \bigl\vert {{b_{i}}(s,x,t)} \bigr\vert \leqslant c{ \vert s \vert ^{1 + \beta }},\qquad \bigl\vert {{{b}_{is}}(s,x,t)} \bigr\vert \leqslant c{ \vert s \vert ^{\beta}}, $$
(1.11)
and
\(u_{0}\)
satisfies
$$ {u_{0}} \in {L^{\infty}}(\Omega),\quad {d^{\alpha}} \vert \nabla {u_{0}} \vert ^{p^{+}} \in {L^{1}}(\Omega), $$
(1.12)
then equation (1.4) with initial value (1.5) has a solution.
We can prove Theorem 1.3 in a similar way to Theorem 1.2 in [11], though Definition 1.2 here is different from that of the weak solution in [11]. We omit the details of the proof here.
In our paper, we will prove another existence result, which seems more interesting.
Theorem 1.4
Let
\(b_{i}(s,x,t)\)
be a
\(C^{1}\)
function, \(p(x)\geq 2\). If
\(\vert s \vert \leq c\),
$$ \bigl\vert {{{b}_{is}}(s,x,t)} \bigr\vert \leqslant cd^{\frac{\alpha}{p(x)}} $$
(1.13)
and
$$ {u_{0}} \in {L^{\infty}}(\Omega),\quad d^{\alpha}{ \vert {\nabla u_{0}} \vert ^{p(x)}} \in {L^{\infty}}\bigl(0,T;L^{1}(\Omega)\bigr), $$
(1.14)
then there is a solution of equation (1.4) with the initial value (1.5).
One can see that only if \(\alpha>0\) in Theorem 1.4 is required, while \(0<\alpha<\frac{p^{-}-2}{2}\) in Theorem 1.3 has a stronger restriction. Moreover, there is a difference between the condition (1.11) and the condition (1.13). As we had shown in [11] only if \(\alpha< p^{-}-1\), the usual Dirichlet boundary condition
$$ u(x,t)=0, \quad (x,t)\in \partial \Omega\times (0,T), $$
(1.15)
can be imposed, and by the condition (1.11), \(b_{i}(0,x,t)=0\). Accordingly, the stability of the weak solutions can be proved. Instead of (1.11), the condition (1.13) has the degeneracy on the boundary independent of the boundary value condition.
The most significant result of our paper is the following stability theorems.
Theorem 1.5
Let
\(u,v\)
be two solutions of (1.4) with the initial values
\(u_{0}(x), v_{0}(x)\), respectively. If
\(b_{i}(s,x,t)\)
satisfies
$$ \bigl\vert b_{i}(u,x,t)-b_{i}(v,x,t) \bigr\vert \leq c d^{\frac{\alpha}{p(x)}} \vert u-v \vert ^{1+\frac{1}{q(x)}}, $$
(1.16)
and the constant
α
satisfies
$$ n \bigl\Vert d^{\alpha-1+\frac{\alpha}{p(x)}} \bigr\Vert _{L^{p(x)}(\Omega\setminus \Omega_{\frac{1}{n}})}\leq c, $$
(1.17)
then
$$ \int_{\Omega} \bigl\vert u(x,t)-v(x,t) \bigr\vert \,dx\leq \int_{\Omega} \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert \,dx. $$
(1.18)
Here
\(\Omega_{\frac{1}{n}}=\{x\in \Omega: d(x)>\frac{1}{n}\}\).
Theorem 1.6
Let
\(u,v\)
be two solutions of (1.4) with the initial values
\(u_{0}(x), v_{0}(x)\), respectively, and
$$ n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}}d^{\alpha} \vert \nabla u \vert ^{p(x)}\,dx \biggr)^{\frac{1}{q^{+}}}\leq c,\qquad n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}}d^{\alpha} \vert \nabla v \vert ^{p(x)}\,dx \biggr)^{\frac{1}{q^{+}}}\leq c. $$
(1.19)
If
\(b_{i}(s,x,t)\)
satisfies (1.16), then the global stability (1.18) is true. Here, \(q(x)=\frac{p(x)}{p(x)-1}\), \(q^{+}=\max_{x\in \overline{\Omega}}q(x)\).
One can see that, in Theorem 1.6, the stability is obtained only in the kind of weak solutions which satisfy the condition (1.19). While the restriction in Theorem 1.5 is the condition (1.17), no restrictions are imposed on the solutions themselves. At the end of the introduction, let us give two sufficient conditions of the condition (1.17).
If (1.17) is true, then \(\Vert d^{\alpha-1+\frac{\alpha}{p(x)}} \Vert _{L^{p(x)}(\Omega\setminus \Omega_{\frac{1}{n}})}<1\), thus by (iii) of Lemma 1.1,
$$ \begin{aligned}[b] n \bigl\Vert d^{\alpha-1+\frac{\alpha}{p(x)}} \bigr\Vert _{L^{p(x)}(\Omega\setminus \Omega_{\frac{1}{n}})} &\leq n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} d^{(\alpha-1+\frac{\alpha}{p(x)})p(x)}\,dx \biggr)^{\frac{1}{p^{+}}} \\ &=n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} d^{p(x)(\alpha-1)+\alpha}\,dx \biggr)^{\frac{1}{p^{+}}} \\ &=n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} d^{\alpha[p(x)(1-\frac{1}{\alpha})+1]}\,dx \biggr)^{\frac{1}{p^{+}}}. \end{aligned} $$
(1.20)
If \(\alpha\geq \frac{p^{+}}{p^{-}+1}\geq \frac{p(x)}{p(x)+1}\), then \(p(x)(1-\frac{1}{\alpha})+1\geq 0\), and
$$ \begin{aligned}[b] n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} d^{\alpha[p(x)(1-\frac{1}{\alpha})+1]}\,dx \biggr)^{\frac{1}{p^{+}}} &\leq n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} d^{\alpha[p^{+}(1-\frac{1}{\alpha})+1]}\,dx \biggr)^{\frac{1}{p^{+}}} \\ &\leq \biggl(\frac{1}{n} \biggr)^{[p^{+}(1-\frac{1}{\alpha})+2]\frac{1}{p^{+}}-1}, \end{aligned} $$
(1.21)
which goes to zero as n goes the infinity, provided that \(\alpha\geq \frac{p^{+}}{2}\), which implies that
$$\biggl[p^{+}\biggl(1-\frac{1}{\alpha}\biggr)+2\biggr] \frac{1}{p^{+}}-1\geq 0. $$
Thus, the condition
$$ \alpha\geq \max\biggl\{ \frac{p^{+}}{p^{-}+1},\frac{p^{+}}{2}\biggr\} , $$
(1.22)
is a sufficient condition of (1.17).
If
$$ \frac{p(x)}{p(x)+2}< \alpha< \frac{p(x)}{p(x)+1}, $$
(1.23)
then
$$\begin{aligned} &{-}1< p(x) \biggl(1-\frac{1}{\alpha}\biggr)+1< 0, \\ &\begin{aligned}[b] n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} d^{\alpha[p(x)(1-\frac{1}{\alpha})+1]}\,dx \biggr)^{\frac{1}{p^{+}}} &=n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} \biggl(\frac{1}{d^{\alpha}}\biggr)^{-[p(x)(1-\frac{1}{\alpha})+1]}\,dx \biggr)^{\frac{1}{p^{+}}} \\ &\leq n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} \biggl(\frac{1}{d^{\alpha}}\biggr)^{p^{+}(\frac{1}{\alpha}-1)-1}\,dx \biggr)^{\frac{1}{p^{+}}} \leq \biggl(\frac{1}{n} \biggr)^{\frac{1}{\alpha}-2}. \end{aligned} \end{aligned}$$
(1.24)
Thus, if
$$ \frac{p^{+}}{p^{-}+2}< \alpha< \min\biggl\{ \frac{p^{-}}{p^{+}+1}, \frac{1}{2}\biggr\} , $$
(1.25)
then \(\alpha\leq \frac{1}{2}\) and \(-1< p(x)(1-\frac{1}{\alpha})+1<0\), so
$$n \biggl( \int_{\Omega\setminus \Omega_{\frac{1}{n}}} d^{\alpha[p(x)(1-\frac{1}{\alpha})+1]}\,dx \biggr)^{\frac{1}{p^{+}}} \leq \biggl(\frac{1}{n} \biggr)^{\frac{1}{\alpha}-2}\leq c. $$
Consequently, the condition (1.25) is another sufficient condition of (1.17).
The paper is arranged as follows. In the first section, we have introduced the basic background and the main results. In the second section, the existence of the weak solution is proved. In the third section, the stability results are obtained. In the last section, we will give a local stability of the weak solutions, without the restriction (1.17).