Regularity of pullback attractors for nonautonomous nonclassical diffusion equations
The authors consider the nonautonomus nonclassical reaction-diffusion equation \[ \begin{aligned} u_{t}-\Delta u_t -\Delta u+f(u) & =g(t) \quad (r,\infty) \times \Omega, \\ u|_{\partial \Omega} & = 0\quad [r,\infty) \times\partial \Omega,\\ u|_{t=0} & =u_r \quad \{0\} \times \Omega,\end{aligned} \] in a bounded domain \(\Omega\subset\mathbb{R}^3\), where \(u_r\in H_0^1(\Omega)\). The nonlinearity \(f\) is continuous, \(f(0)=0\) and it satisfies the growth and dissipativity conditions \[ \begin{aligned} |f(u)-f(v)| & \leq c|u-v|(1+|u|^4+|v|^4),\\ \liminf_{|u|\to\infty}\frac{f(u)}{u} & >-\lambda_1, \end{aligned} \] where \(\lambda_1>0\) denotes the first eigenvalue of the Dirichlet Laplacian \(A:=-\Delta\) in \(\Omega\). Moreover, \(g\) satisfies \(g\in L^2_{\mathrm{loc}}(\mathbb{R};H^{-1}(\Omega))\) and \[ \sup_{r\leq t}\left(\int_{r-1}^r \|g(s)\|^2_{\mathcal{E}^{-\frac{2}{3}}}ds\right)<\infty, \] where \(\mathcal{E}^{-\frac{2}{3}}=D(A^{-\frac{1}{3}})\) denotes the fractional power space associated to \(A\). The existence of a pullback attractor in \(H_0^1(\Omega)\) is established and, under certain additional assumptions on \(g\), the boundedness of the pullback attractor in \(H^1_0(\Omega)\cap H^2(\Omega)\) is shown.
scientific article; zbMATH DE number 6817529
scientific article