JOURNAL OF FUNCTIONAL ANALYSIS 71. 142-174 (1987) Complete Blow-Up after T,,, for the Solution of a Semilinear Heat Equation P. BARAS L. rhoratoire TIM-3-IMAG. B.P. 68. 38401 Saint Martin D’Hrrcs Ceder. Frunw AND L. COHEN 32, Rue du Jawlot, 75645 Park Cede.\- 13. France Conmunicated by, H. Brcis Received June 1985; revised December 1985 INTRODUCTION Let !2 be a bounded open subset of [WV with a smooth boundary X?. Consid :r the problem u, - Au =f’( u) in Q x (0, T), u=o on ?R x (0, T), (PI u( A-, 0) = u,(x) for all s E R. where “‘i lR+ + R+ is locally Lipschitz, nondecreasing and f’(0) = 0. If liO is a continuous function on ~7. there exists a unique classical solution u of (P) defined on [0, T,,,) and such that u E %“,‘(Q x (0, T,,,)) n %‘(Q x 110, T,,,)) with lim,, r,,,, Ilull % = ~j if T,,,, < X. A well-known result asserts that if u is large enough and f(u) = u”, p > 1, for example, then T,,, < SC) (this is the case when l/2 IVu,l’ - l/(p + I ) jn luOl pf ’ < 0 see. for example, [ 1] or Corollary 2.2). In what follows, we suppose that T,,, < + 8~. Assu ne ,f,, : R + -+ R + is a sequence of functions such that for each n, II -f;,(u) is globally Lipschitz, non decreasing, .f;,co,(f 3. (b for each U, n +.f,(u) is increasing and converges to .f(u). 142 0022- 123 i;87 $3.00 Copyright t 1987 by Academic Press. [nc All nghk 01 reproductmn m any form reserted.