ON THE CAUCHY-PROBLEM FOR REACTION-DIFFUSION EQUATIONS

The simplest model of the Cauchy problem considered in this paper is the following u(t) = DELTAu + u(p), x is-an-element-of R(n), t > 0, u greater-than-or-equal-to 0, p > 1, u\t = 0 = phi is-an-element-of C(B)(R(n)), phi greater-than-or-equal-to 0, phi not-equal 0. It is well known that when 1 < p less-than-or-equal-to (n + 2)/n, the local solution of blows up in finite time as long as the initial value phi is nontrivial; and when p > (n + 2)/n , if phi is ''small'', (*) has a global classical solution decaying to zero as t --> +infinity, while if phi is ''large'', the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on phi for global existence and to study the asymptotic behavior of those global solutions. In particular, we prove that if n greater-than-or-equal-to 3, p > n/(n - 2), 0 less-than-or-equal-to phi(x) less-than-or-equal-to lambdau(s)(x) = lambda(2(n - 2)/(p - 1)2 (p - n/n - 2))1/(p - 1) Absolute value of x -2/(p - 1) (u(s) is a singular equilibrium of (*)) where 0 < lambda < 1 , then (*) has a (unique) global classical solution u with 0 less-than-or-equal-to u less-than-or-equal-to lambdau(s) and u(x, t) less-than-or-equal-to ((lambda1 - p - 1)(p - 1)t) - 1/(p - 1). (This result implies that u0 = 0 is stable w.r.t. to a weighted L(infinity) topology when n greater-than-or-equal-to 3 and p > n/(n - 2).) We also obtain some sufficient conditions on phi for global nonexistence and those conditions, when combined with our global existence result, indicate that for phi around u(s), we are in a delicate situation, and when p is fixed, u0 = 0 is 'increasingly stable'' as the dimension n up +infinity. A slightly more general version of (*) is also considered and similar results are obtained.