A necessary and sufficient condition for the global existence of solutions to nonlinear reaction-diffusion equations on the half-spaces in R<SUP>N</SUP>

초록

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction-diffusion equations {u(t)(x, t) = Delta u(x, t) + psi (t) (u(x, t)), (x, t) is an element of Omega x(0, infinity), u(center dot, 0) = u(0)(x), x is an element of Omega, u(x, t) = 0, (x, t) is an element of partial derivative Omega x (0, infinity), where Omega is the half-space R-K(N), psi is a nonnegative continuous function, and.. is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and non-trivial initial data u(0) is an element of C-0(Omega) if and only if integral(infinity)(1) psi(t)t(N+K/2) integral (epsilon t-(N+K/2)) dt = infinity for every epsilon > 0. In fact, we introduce a very special curve in R-K(N) (x) over cap (t) := {root t, center dot center dot center dot, root t,((sic)K-times) x(K+1), x(N)}, t > 0, to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result.

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