LOWER AND UPPER SOLUTIONS FOR THE HEAT EQUATION ON A POLYGONAL DOMAIN OF R-2

作者：De Coster Colette^*; Nicaise Serge

来源：Differential and Integral Equations, 2013, 26(5-6): 603-622.

We consider the nonlinear periodic-Dirichlet heat equation on a polygonal domain Omega of the plane in weighted L-p-Sobolev spaces %26lt;br%26gt;partial derivative(t)u - Delta u = f(x, t, u), in Omega x (-pi, pi), %26lt;br%26gt;u = 0, on partial derivative Omega x (-pi, pi), %26lt;br%26gt;u(., -pi) = u(., pi) in Omega. %26lt;br%26gt;Here f is L-p(0, T; L-mu(p) (Omega)-Caratheodory, where L-mu(p)(Omega) = {v is an element of L-loc(p)(Omega) : r(mu) v is an element of L-p(Omega)}, with a real parameter mu and r(x) the distance from x to the set of corners of Omega. We prove some existence results of this problem in presence of lower and upper solutions well-ordered or not. We first give existence results in an abstract setting obtained using degree theory. We secondly apply them for polygonal domains of the plane under geometrical constraints.

• 出版日期2013-6