Global Existence And Blow-up Profiles For The Solutions Of Some Reaction-diffusion Equations

Reaction-diffusion equations come from many mathematical models in physics, chemistry and biology, which have strongly practical background; on the other hand, in the studying of reaction-diffusion equation, many important problems are developed. In the recent twenties years, reaction-diffusion equations are investigated by more and more mathematicians, physicists, chemists, biologists and engineers. This paper deals with global existence and blow-up profiles for the solutions of some classes of reaction-diffusion equations.In chapter 1, we deal with the Cauchy problem for weakly coupled reaction-diffusion system u_t — Δu = t~μ|x|~mv~p, v_t — Δv = t~σ|x|~nu~q. By the technique of analysis and iteration, we get the Fujita-type blow-up critical exponent. Precisely, let γ = max{p+ (np/2) + m/2 + σp + μ, q+(mq/2) + n/2 + μq + σ}. If 1 < pq < 1 + 2(1 + γ)/N, all nontrivial solutions blow up at the finite time; If pq > 1 + 2(1 + γ)/N, the solutions blow up at the finite time for large initial data, while there exists global solution for small initial data if we further assume that p, q > 1. These results show the relation between the exponents of spacial and time variables and the Fujita-type blow-up critical exponent.In chapter 2, we deal with the blow-up properties and asymptotic behavior of solutions to a semilinear integrodifferential system with nonlocal reaction terms in space and time. The blow-up conditions are given by a variant of the eigenfunction method combined with new properties on systems of differential inequalities. At the same...