In this paper, we consider an initial boundary value problem for the semilinear parabolic equation with variable exponent:
$$ \textstyle\begin{cases} u_{t}=\Delta {u}+\vert u\vert ^{p(x)-2}u, & x \in \Omega, t>0, \\ u(x,t)=0,& x \in \partial \Omega, t>0, \\ u(x,0)=u_{0}(x), & x\in \Omega, \end{cases} $$
(1.1)
where Ω is a bounded smooth domain of \(\mathbb{R}^{N}\) (\(N \geq 3\)), \(u_{0}\in H_{0}^{1}(\Omega)\), and \(p(x)\) is a continuous and bounded function satisfying
$$ 2< p^{-}:=\inf_{x\in \Omega }p(x)\le p^{+}:=\sup_{x\in \Omega }p(x)< 2^{*}= \frac{2N}{N-2}. $$
(1.2)
Eq. (1.1) has been used to model a variety of important physical processes, such as electrorheological fluids (where u is the velocity of moving fluids in electro-magnetic fields) [1], thermo-rheological flows or population dynamics [2, 3]. There is a substantial amount of work concerning the case \(p(x)\equiv p\), see, for example, [4–6].
To deal with the variable source, it is convenient to introduce a Lebesgue space \(L^{p(\cdot)}(\Omega)\), defined as the space of measurable functions u in Ω satisfying \(\int_{\Omega }\vert u\vert ^{p(x)}\,dx< \infty \). We mention that this kind of Lebesgue space or general Sobolev space with variable exponent and their applications have got a lot of attention, see the monograph [7] and some recent work [8–11] for instance.
With the norm
$$\Vert u\Vert _{p(\cdot)}:=\Vert u\Vert _{L^{p(\cdot)}}(\Omega) =\inf \biggl\{ \lambda >0: \int_{\Omega } \biggl\vert \frac{u}{\lambda } \biggr\vert ^{p(x)}\,dx\le 1 \biggr\} , $$
the space \(L^{p(\cdot)}(\Omega)\) is a Banach space and
$$ \inf \bigl\{ \Vert u\Vert _{p(\cdot)}^{p^{+}}, \Vert u\Vert _{p(\cdot)}^{p^{-}} \bigr\} \leq \int_{\Omega }\vert u\vert ^{p(x)}\,dx \leq \max \bigl\{ \Vert u\Vert _{p(\cdot)}^{p^{+}}, \Vert u\Vert _{p(\cdot)}^{p^{-}}\bigr\} , $$
(1.3)
see [12]. Combining Corollary 3.34 in [12] and the Poincaré inequality, we have
$$\begin{aligned} \Vert u\Vert _{p(\cdot)}\le B \Vert \nabla u\Vert _{2}. \end{aligned}$$
(1.4)
Regarding variable sources, Pinasco [13] proved that the solution of (1.1) blows up in finite time provided that \(p^{-}>1\) and the initial data is large enough. This result was then extended to \(p^{+}>1\) by Ferreira et al. in [14]. For some positive initial energy, Wu et al. [15] gave a blow-up condition.
Proposition 1.1
Theorem 1.1 in [15]
Let
$$ E_{1}=:\frac{1}{p^{-}} \biggl(\frac{p^{+}-2}{2}B^{p^{+}} \alpha_{1}^{\frac{p ^{+}}{2}} +\frac{p^{-}-2}{2}B^{p^{-}} \alpha_{1}^{\frac{p^{-}}{2}} \biggr), $$
(1.5)
and
$$\overline{E}_{1}=: \biggl(\frac{p^{+}-2}{p^{-}-2} \biggr)^{\frac{2}{p^{+}}} \biggl\{ \frac{\alpha_{1}}{2}-\frac{1}{p^{-}} \biggl[B^{p^{+}} \biggl( \frac{p ^{+}-2}{p^{-}-2} \biggr)^{\frac{p^{+}-2}{p^{+}}}\alpha_{1}^{ \frac{p^{+}}{2}} +B^{p^{-}} \biggl(\frac{p^{+}-2}{p^{-}-2} \biggr)^{\frac{p ^{-}-2}{p^{-}}} \alpha_{1}^{\frac{p^{-}}{2}} \biggr] \biggr\} < E_{1}, $$
where
\(\alpha_{1}\)
is defined by
$$ \frac{1}{p^{-}} \bigl(B^{p^{+}}p^{+} \alpha_{1}^{\frac{p^{+}-2}{2}}+B ^{p^{-}}p^{-} \alpha_{1}^{\frac{p^{-}-2}{2}} \bigr)=1. $$
(1.6)
Assume
\(1<\sqrt{2p^{+}-1}<p^{-}\le p^{+}\le \frac{N+2}{N-2}\)
and
\(0< E(u_{0})<\overline{E}_{1}\). If
\(\Vert \nabla u_{0}\Vert _{2}^{2}>\alpha _{1}\), then the solution of Eq. (1.1) blows up in finite time.
Later, another blow-up condition was derived by Wang and He [16].
Proposition 1.2
Theorem 1 in [16]
Assume
\(1< p^{-}\le p^{+}\le \frac{N+2}{N-2}\)
and
\(0< E(u_{0})< E_{2}=\frac{p ^{-}-2}{2p^{-}}B_{1}^{-\frac{2p^{-}}{p^{-}-2}}\)
with
\(B_{1}\geq \max \{B,1\}\). If
\(\Vert \nabla u_{0}\Vert _{2}^{2}>\alpha_{2}=B_{1}^{-\frac{2p ^{-}}{p^{-}-2}}\), then the solution of Eq. (1.1) blows up in finite time.
Motivated by the above research, in this paper we have the main purpose to look for a sharp threshold for blow-up and global existence of solutions of (1.1) in general case (1.2) and (1.7) (in Section 4, we show \(E_{0}> \overline{E}_{1}\) and \(E_{0}\geq E_{2}\)). We mainly use the potential well method, which was used to study the case \(p(x)\equiv p\) by Payne and Sattinger [17], and was widely used to consider other parabolic models during the last years, see, for example, [18–21]. Similar to [22], local existence and uniqueness of solutions of (1.1) can be obtained by the Banach fixed point theorem as follows.
Proposition 1.3
Assume that (1.2) holds. Then (1.1) admits a unique solution
\(u\in C ([0,T_{\max });H_{0}^{1}(\Omega)) \cap C^{1} ((0,T_{\max });L^{2}(\Omega))\), where
\(T_{\max }>0\)
denotes the maximal existence time. Either
\(T_{\max }<+\infty \)
and
\(\lim_{t\rightarrow T_{\max }}\Vert u\Vert _{H_{0}^{1}(\Omega)}^{2}=+\infty \) (we say that the solution blows up in finite time), or
\(T_{\max }=+ \infty \) (we say that the solution is global in time).
Denote the energy functional
$$E(u)= \int_{\Omega } \biggl[\frac{1}{2}\bigl\vert \nabla u(x,t) \bigr\vert ^{2}-\frac{1}{p(x)}\bigl\vert u(x,t)\bigr\vert ^{p(x)} \biggr]\,dx $$
and the Nehari manifold
$$\mathcal{N}=\bigl\{ u\in H_{0}^{1}(\Omega)\vert N(u)=0, u \neq 0\bigr\} $$
with
$$N(u)=\bigl\langle E'(u),u\bigr\rangle = \int_{\Omega } \bigl[\bigl\vert \nabla u(x,t)\bigr\vert ^{2}-\bigl\vert u(x,t)\bigr\vert ^{p(x)} \bigr]\,dx. $$
In this paper, we assume that the initial energy is less than the potential well depth, namely
$$\begin{aligned} E(u_{0})< E_{0}:=\inf_{u\in \mathcal{N}}E(u). \end{aligned}$$
(1.7)
Now, we introduce our main results as follows.
Theorem 1.1
Assume that (1.2) and (1.7) hold. Then
-
(1)
if
\(N(u_{0})<0\), then the solution of Eq. (1.1) blows up in finite time;
-
(2)
if
\(N(u_{0})\geq 0\), then the solution
u
of Eq. (1.1) is global in time and
\(u(t)\rightarrow 0\)
strongly in
\(H^{1}_{0}(\Omega)\)
as
\(t\rightarrow \infty \).
Finally, we consider applications of Theorem 1.1 and derive the following results.
Corollary 1.1
Assume (1.2) and
\(0< E(u_{0})<\overline{E}_{1}\). The solution to Eq. (1.1) is global in time if
\(\Vert \nabla u_{0}\Vert _{2}^{2} \leq \alpha_{1}\).
Corollary 1.2
Assume (1.2) and
\(0< E(u_{0})< E_{2}\). The solution of Eq. (1.1) is global in time if
\(\Vert \nabla u_{0}\Vert _{2}^{2} \leq \alpha_{2}\).
Remark 1.1
Combined with Proposition 1.1 and Proposition 1.2, the above corollaries imply that the blow-up conditions in [15] and [16] are also sharp there.
This paper is organized as follows. In Section 2, we determine the blow-up condition of solutions of Eq. (1.1). In Section 3, we deal with global existence condition and then conclude that the global solution decays as the time goes to infinity. In Section 4, we prove Corollaries 1.1 and 1.2. Finally, we summarize the main results of the current paper.
In the sequel, we use \(\Vert \cdot \Vert _{p}\) to denote \(L^{p}(\Omega)\) norm, and denote the inner product in \(L^{2}(\Omega)\) by the symbol \((\cdot ,\cdot )\).