The Nature Of The Solution Of Two Class Of Reaction-Diffusion Equations

In this paper, we study the mixed problem for the following fast diffusion systems with nonlinear sources. Where 0< m< 1 and a, b,> 0, Ω be a bounded domain in RN (N> 2),with smooth boundary (?)Ω,u0,vo are nonnegative L∞(Q) functions. It is show that the system is singular and there is no classical solution in general, so we give the definition of weak solution. At last, we give some sufficient conditions for the solutions of fast diffusion systems to vanish in finite time, by using the theory of invariant region for ODE systems and integral estimates.And we investigate quenching behavior of solutions in coupled Reaction-Diffusion Systems with singular nonlinearities. Where m, n≥0, p, q> 0, α, β> 0 and the initial data u0, v0 satisfyingIn the initial boundary conditions, we study the solution of equation of quenching, quenching point set, and quenching time derivative in the asymptotic behavior of quenching point by using the maximum principle. Finally, we get a quenching rate by using a comparison lemma.