A note on well-posedness of semilinear reaction-diffusion problem with singular initial data

作者：Robinson James C; Sierzega Mikolaj^*

来源：Journal of Mathematical Analysis and Applications, 2012, 385(1): 105-110.

DOI：10.1016/j.jmaa.2011.06.023

• Reaction-diffusion equation
• Singular initial conditions
• Well-posedness

We discuss conditions for well-posedness of the scalar reaction-diffusion equation u(t) = Delta u + f(u) equipped with Dirichlet boundary conditions where the initial data is unbounded. Standard growth conditions are juxtaposed with the no-blow-up condition integral(infinity)(1) 1/integral (s) = infinity that guarantees global solutions for the related ODE (u) over dot = f (u). We investigate well-posedness of the toy PDE u(t) = f(u) in L(P) under this no-blow-up condition. An example is given of a source term f and an initial condition Psi epsilon L(2)(0, 1) such that integral(infinity)(1) 1/f (s) ds = infinity and the toy PDE blows-up instantaneously while the reaction-diffusion equation is globally well-posed in L(2)(0,1).

• 出版日期2012-1-1