ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY

We consider the growth rate and quenching rate of the following problem with singular nonlinearity ut = Delta u - upsilon(t) =Delta nu -u(-u,) (x,t) is an element of R-n x (0,infinity), u(x, 0) = u(o)(x), v(x, 0) = v(o) (x), x is an element of R-n for any n >= 1, where lambda,mu > 0 are constants. More precisely, for any u(o)(x), v(o)(x) satisfying A(11)(1+vertical bar x vertical bar(2))(alpha 11) <= u(o) <= A(12)(1+vertical bar x vertical bar(2))(alpha 12), A(21)(1+ vertical bar x vertical bar(2))(alpha 12) <= v(o) <= A(22)(1+ vertical bar x vertical bar(2))(alpha 22) for some constants alpha(12) >= alpha(11), alpha(22) >= alpha(21), A(12) >= A(11), A(22) >= A(21), the global solution (u, v) exists and satisfies Aii(1+1x12 +bit)an <U < Al2(1+1x12+b2t)a12, A21 (1 IX12 bit)n21 G. V G A(22) (1+ vertical bar x vertical bar(2) +b(2)t)(alpha 22) for some positive constants b(1), b(2) (see Theorem 3.3 for the parameters Ath cx,b,, j i=x1, 2). When (1- A)(1 - > 0, (1- A)(1 -)A) > 0 and 0 < uo Ai (biT +vertical bar x vertical bar(2)) 1 A1', 0 < VO < A(2) (b(2)T + I X12) in Rn for some constants At, b (i = 1,2) satisfying A(2)-A > 2nA1 A > 2nA2 and 0 < b(1) < (1)p,),4 (1 A)2nAi 0 < b(2) < (1- 2nA2 we prove that u(x, t) < A(i) (b(i) (T -t) lx12)1-, v(x, t) < A(2)(b(2)(T - t) vertical bar x vertical bar(2))1-1-1-, in Rn X (0, T). Hence, the solution (u, v) quenches at the origin x = 0 at the same time T (see Theorem 4.3). We also find various other conditions for the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.