Periodic solutions of evolution \(p\)-Laplacian equations with nonlinear sources
Time-periodic solutions to the \(p\)-Laplacian evolution equations \[ \partial u/\partial t= \text{div} (|\nabla u |^{p-2}\nabla u)+ m( x, t) u^\alpha \quad \text{in } \Omega\times \mathbb{R}\tag{1} \] and \[ \partial u/\partial t= \sum_{i=1}^N \partial/\partial x_i (|\partial u/\partial x_i|^{p-2}\partial u/\partial x_i)+ m(x,t)u^\alpha \quad \text{in } \Omega \times \mathbb{R}\tag{2} \] are sought on a bounded convex domain \(\Omega \subset \mathbb{R}^N\), for \(N\), and \(p \geq 2\), zero Dirichlet conditions being required on the smooth boundary \( \partial \Omega \). The source term \(m(x,t)\) is assumed to be positive, continuous on \(\bar \Omega \times \mathbb{R} \) and to be \(t\)-periodic, with some period \(\omega > 0\). For the case of strongly nonlinear sources, namely \(\alpha > p-1\), a blow-up (scaling) argument is employed to deduce the existence of nonnegative continuous periodic solutions to (1), provided \(p,\alpha + 1 < p ( 1 + 1/N)\). The same argument applies also to (2), the solution in the former case satisfying \(u \in C(0, \omega; W_0^{1,p}(\Omega)) \cap C_\omega (\overline {Q})\), \(\partial u /\partial t \in L^2(Q)\) while in the latter it is \(u \in C(0, \omega; W_0^{1,0}(\Omega)) \cap C_\omega (\bar Q)\), \(\partial u /\partial t \in L^2(Q)\) which holds. Observe that due to the degeneracy of the equations being considered, the solutions exist in a weak sense with respect to the time variable.
nonnegative continuous periodic solutions
scientific article; zbMATH DE number 1218657
scientific article