×

On the antimaximum principle for parabolic periodic problems with weight. (English) Zbl 1098.35069

It is proved that an antimaximum principle (AMP) holds for the Dirichlet and Neumann parabolic linear problems of second order of the type \(Lu=\lambda mu+h\) in \(\Omega \times \mathbb R\), where \(\Omega \subset \mathbb R^{N}\) is a bounded domain, the coefficients of \(L\) are sufficiently smooth functions, \(\tau\)-periodic in \(t\), the weight function \(m(x,t)\) is \(\tau\)-periodic in \(t\) and bounded, \(h(x,t)\) is \(\tau\)-periodic in \(t\) and \(h\in L^{p}(\Omega \times (0,\tau ))\) for some \(p>N+2.\) Depending on \(m\), it can be proved that in some cases the AMP holds left and right of \(\lambda^\star\), where \(\lambda^\star\) is the principal eigenvalue of the previous problem (i.e. for \(\lambda = \lambda^\star\) and \(h \equiv 0\) the problem has a positive solution).

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35P05 General topics in linear spectral theory for PDEs
47N20 Applications of operator theory to differential and integral equations
35B50 Maximum principles in context of PDEs
PDFBibTeX XMLCite
Full Text: EuDML