A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

作者：Laister, R.; Robinson, J. C.^*; Sierzega, M.; Vidal-Lopez, A.

来源：ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, 2016, 33(6): 1519-1538.

DOI：10.1016/j.anihpc.2015.06.005

• Semilinear heat equation
• Dirichlet problem
• Local existence
• Non-existence
• Instantaneous blow-up
• Dirichlet heat kernel

We consider the scalar semilinear heat equation u(t) - Delta u = f (u), where f : [0, infinity) -> [0, infinity) is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in L-q (Omega) for all non-negative initial data u(0) epsilon L-q (Omega), when Omega subset of R-d is a bounded domain with Dirichlet boundary conditions. For q epsilon (1, infinity) this holds if and only if lim sup(s ->infinity)s(-(1+ 2q/d)) f (s) < infinity; and for q =1 if and only if integral(infinity)(1) s(-(1+2/d)) F(s) ds < infinity, where F(s) = sup1 <= t <= s f (t)/t. This shows for the first time that the model nonlinearity f (u) = u(1+2q/d) is truly the 'boundary case' when q epsilon (1, infinity), but that this is not true for q = 1. The same characterisations hold for the equation posed on the whole space R-d provided that lim sup(s -> 0) f (s)/s < infinity.

• 出版日期2016-12
• 单位西交利物浦大学