Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity
The authors investigate the existence and stability of \(T\)-periodic solutions of NEWLINE\[NEWLINEu_t-k\partial_t \Delta u=m(t)u\log(\left|u\right|),\quad x\in\Omega,\;t\in\mathbb{R}; \qquad u(x,t)=0, \quad x\in\partial\Omega,\;t\in\mathbb{R}NEWLINE\]NEWLINE under the following assumptions: \(T>0\), \(k\in\{0,1\}\) (\(k=1\) in pseudo-parabolic case), \(\Omega\subset \mathbb{R}^n\) bounded domain with smooth boundary, \(m\) is a smooth, positive and T -periodic function. They find that existence results are similar to those for right hand sides \(m(t)u^p\), \(p>1\), whereas there are differences between the instability results for the linear case, the `logarithmic case', and the `power case'. The existence proofs rely on regularity results, a priori estimates, and topological degree arguments; the stability results use the comparison method.
scientific article; zbMATH DE number 6639797
scientific article