ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 63,   2   (1994)
pp.   169-192

ON CRITICAL EXPONENTS FOR A SYSTEM OF HEAT EQUATIONS COUPLED IN THE BOUNDARY CONDITIONS
K. DENG, M. FILA and H. A. LEVINE

Abstract.  In this paper, we consider the system \alignat 4 u_t& = \Delta u,& \qquad v_t&= \Delta v &\qquad &x\in \Bbb R_+^N, &\qquad t &> 0, \ -\frac\partial u\partial x_1& = v^p,& \qquad-\frac\partial v\partial x_1& = u^q &\qquad &x_1 = 0, &\qquad t & > 0, \ u(x,0) &= u_0(x),&\qquad v(x,0) &= v_0(x) &\qquad &x\in \Bbb R_+^N, &\qquad & \endalignat where $\Bbb R_+^N = \(x_1,x') \vert x' \in \Bbb R^N-1, x_1 > 0\$, $p, q > 0$, and $u_0$, $v_0$ nonnegative. We prove that if $pq \le 1$ every nonnegative solution is global. When $pq > 1$ we let $\a=\frac 12\fracp+1pq-1$, $\b= \frac 12\frac q+1pq-1$. We show that if $\max (\a,\b)\ge \frac N2$, all nontrivial nonnegative solutions are nonglobal; whereas if $\max (\a,\b)< \frac N2$ there exist both global and nonglobal nonnegative solutions. When $N=1$, we establish some results for the blow up rate for the nonglobal solutions and some results for the decay rate for the global solutions (in the supercritical case). We also construct a nontrivial solution with vanishing initial values when $pq<1$.

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