Existence and stability of bounded solutions for a system of parabolic equations
In this paper the authors discuss the existence and the stability of bounded solutions of the following nonlinear system of parabolic equations with homogeneous Dirichlet boundary conditions: \[ \begin{cases} u_t = D\Delta u + f(t,u), & \text{in }\Omega \quad t \geq 0, \;u \in \mathbb{R}^n, \\ u=0 & \text{on }\partial \Omega. \end{cases} \] where \(f \in C^1\), \(D=\text{diag}(d_1,d_2,\dots,d_n)\) is a diagonal matrix with \(d_j>0\), \(i=1,2,\ldots, n\), and \( \Omega\) is a sufficiently regular bounded domain in \(\mathbb{R}^N (N = 1, 2, 3)\). The authors prove the following results: If \(f\) is globally Lipschitz with constant \(L\), \[ {3 \over 4} < \alpha < 1 \quad \text{and} \quad {(\lambda_1d)^{1-\alpha} \over \Gamma(1-\alpha)} > 6ML, \] then the system has a bounded solution which is stable, where \(2d = \min \{d_i: i=1,2,\ldots, n\}\), \((\lambda_jd_it)^\alpha e^{-\lambda_j(d_i/2)t}<M\), \(\lambda_i\) are the eigenvalues of \(-\Delta\), and \(\Gamma(\cdot)\) is the well-known gamma function. Also, the authors prove that for a large class of functions \(f\) this bounded solution is almost periodic.
homogeneous Dirichlet boundary conditions
scientific article; zbMATH DE number 1915765