Long-time behavior for semilinear degenerate parabolic equations on \(\mathbb R^N\) (Q2868488)

The authors considered the following semilinear degenerate parabolic equation: NEWLINE\[NEWLINE\frac{\partial u}{\partial t}-\text{div}\;\left(\sigma(x)\;\nabla u\right)+\lambda u +f(u)=g(x)\;\text{in}\;\mathbb{R}^N \times (0,\infty)NEWLINE\]NEWLINE under the initial condition \(u(x,0)=u_0(x)\), where \(\lambda>0\), \(u_0\), \(g \in L^{2}(\mathbb{R}^{N})\), and \(f \in C^{1}(\mathbb{R};\mathbb{R})\) satisfies the following conditions: NEWLINE\[NEWLINEf(u) u \geq -\mu u^2,\;\mu < \lambda,\quad f^{'}(u) \geq -C,\quad |f(u)| \leq C\;|u|.NEWLINE\]NEWLINE The diffusion coefficient \(\sigma\) is assumed to be a nonnegative measurable function belonging to \(L_{\text{loc}}^{1}(\mathbb{R})^N\) and such that for some \(\alpha \in (0,2)\) NEWLINE\[NEWLINE\liminf_{x \rightarrow z}|x-z|^{-\alpha}\sigma(x)>0\;\forall z \in \mathbb{R}^N.NEWLINE\]NEWLINE It is also assumed that \(\sigma\) satisfies either NEWLINE\[NEWLINE\exists K_0>0:\; \sup_{k \geq K_0}\sup_{k \leq |x| \leq \sqrt{2}k}\sigma(x)<\inftyNEWLINE\]NEWLINE or NEWLINE\[NEWLINE\exists K_0>0:\;\sup_{k \geq K_0} \int_{k \leq |x| \leq \sqrt{2}k} |\sigma(x)|^{N/(2-\alpha)}\;dx <\infty.NEWLINE\]NEWLINE This new condition on \(\sigma\) ensures the asymptotic compactness of the semigroup generated by the unique weak solution of the original problem and also avoids imposing limiting behavior of \(\sigma\) at infinity. The first main result of the paper is the existence of weak solutions.NEWLINENEWLINENEWLINE{Theorem 1}: Given \(T>0\), there exists a unique weak solution \(u\) on \([0,T]\), and for each \(t \in [0,T]\) the map \(u_0 \mapsto u(t)\) is continuous on \(L^2(\mathbb{R}^N)\).NEWLINENEWLINENEWLINEIn the paper, \(H_{0}^{1}(\mathbb{R}^{N},\sigma)\) represents the closure of \(C_{0}^{\infty}(\mathbb{R}^N)\) with respect to the norm NEWLINE\[NEWLINE\|v\|_{H_{0}^{1}(\mathbb{R}^N,\sigma)}:=\left(\int_{\mathbb{R}^N}\sigma(x)\;|\nabla v|^2\;dx+\int_{\Omega}|v|^2\;dx\right)^{1/2}.NEWLINE\]NEWLINE Using Theorem \(1\), a continuous semigroup \(S(t):L^{2}(\mathbb{R}^N) \rightarrow H_{0}^{1}(\mathbb{R}^{N},\sigma)\) is defined by \(S(t)\;u_0:=u(t)\) and the following Theorem is proved using the tail estimate method originally proposed by \textit{B. Wang} [Physica D 128, No. 1, 41--52 (1999; Zbl 0953.35022)].NEWLINENEWLINE NEWLINE{Theorem 2 }: The semigroup \(S(t)\) has a compact connected global attractor \(\mathcal{A}\) in \(L^{2}(\mathbb{R}^{N})\).NEWLINENEWLINEIn the final section of the paper, under the additional assumption on \(f\) that NEWLINE\[NEWLINEF(u) \geq -\frac{\mu}{2}u^2,\quad \mu<\lambdaNEWLINE\]NEWLINE with \(F(u)=\int_{0}^{u}f(s) ds\), the existence of a global attractor in \(H_{0}^{1}(\mathbb{R}^{N},\sigma)\) is established by proving uniform estimates of the first derivative with respect to time of the solution to show that the semigroup \(S(t)\) is asymptotically compact in \(H_{0}^{1}(\mathbb{R}^{N},\sigma)\).