A Uniqueness Result for a Semilinear Reaction-Diffusion System

Abstract

Abstract. Let (u(t, x), v(t, x)) and (u(t, x), v(t, x)) be two nonnegative classical solutions of {ut = Am + v" , p > 0 v, = Av + uq , q > 0 in some strip ST = (0, T) x R , where 0 < T < oo , and suppose thatwhere «(0, x) and t;(0, x) are continuous, nonnegative, and bounded real functions, one of which is not identically zero. Then one hasIf pq > \ , the result is also true if w(0, x) = v(0, x) = 0 . On the other hand, when 0 < pq < 1 , the set of solutions of (S) with zero initial values is given by u(t;s) = cx(t-sf+ + X)l{X-pQ), v(t;s)=c2(t-s)l:+mi-M), where 0 < s < f, c¡ and c2 are two positive constants depending only on p and q , and (<J), = max{£, 0} .

Large Time Behavior and Uniqueness of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

“…Theorems 1, 2 and 3 in this paper extend these results to the general case N ≥ 3. Basically Theorems 1, 2 and 3 can be proved in a similar way to that in [2] and [3]. But for our big system, the procedure to obtain the key differential inequalities (23) and (24) below becomes very complicated.…”

“…To this end, we first notice that with α 1 as in (3). We also note the relation p i α i+1 + 1 = α i (i ∈ N * ).…”

“…We consider nonnegative solutions of the initial value problem for a weakly coupled system Since the pioneering work of Fujita [6], the blow up and global existence of solutions to weakly coupled semilinear parabolic systems have been studied by several authors ( [2], [3], [4] and [7]). …”

On uniqueness for a semilinear parabolic system coupled in an equation and a boundary condition

“…A nonuniqueness result is obtained by Deng et al in [4] where they studied the system 5) with N 1, p > 0, q > 0, and u 0 , v 0 being nonnegative, smooth, and bounded functions satisfying the compatibility condition. Their results solve the questions of global existence mainly, nevertheless in the dimension N = 1 they also constructed a nontrivial solution with zero initial data if pq < 1.…”

“…It has the same form as in [5] for (1.3). The solutions are unique if pq 1 or if (u 0 , v 0 ) is nontrivial.…”