Qualitative properties of nonnegative solutions for a doubly nonlinear problem with variable exponents
Summary: We consider the Dirichlet initial boundary value problem \(\partial_t u^{m(x)} - \operatorname{div} (| \nabla u|^{p(x, t) - 2} \nabla u) = a (x, t) u^{q(x, t)}\), where the exponents \(p(x, t) > 1\), \(q(x, t) > 0\), and \(m(x) > 0\) are given functions. We assume that \(a(x, t)\) is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if \(\operatorname{ess} \sup p(x, t) - 1 < \operatorname{ess} \inf m(x)\), then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when \(\operatorname{ess} \sup m(x) < \operatorname{ess} \inf p(x, t) - 1\), we get the positivity of solutions for large \(t\). In the second part, we investigate the property of propagation from the initial data. For this purpose, we give a precise estimation of the support of the solution under the conditions that \(\operatorname{ess} \sup m(x) < \operatorname{ess} \inf p( x, t) - 1\) and either \(q(x, t) = m(x)\) or \(a(x, t) \leq 0\) a.e. Finally, we give a uniform localization of the support of solutions for all \(t > 0\), in the case where \(a(x, t) < a_1 < 0\) a.e. and \(\operatorname{ess} \sup q \left(x, t\right) < \operatorname{ess} \inf p(x, t) - 1\).
scientific article; zbMATH DE number 7029286
scientific article