By T Li, Yun-Mei Chen

This article represents the implications initially acquired through S. Lainerman, D. Christodoulou, Y. Choquet-Bruhat, T. Nishida and A. Matsumara at the worldwide lifestyles of classical ideas to the Cauchy challenge with small preliminary facts for nonlinear evolution equations.

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5. 1. Introduction In this section we shall discuss the Cauchy problem for «-dimensional nonlinear heat equations ¿2 02 ------- + . . 4) 37 NONLINEAR HEAT EQUATIONS and e >0 is a small parameter. ,«). 1) we give the following hypotheses. 6) where a is an integer > 1. 14), provided e > 0 is suitably small. 2) admits a unique global classical solution on r > 0 and this solution possesses the same decay properties when i -> + oo as the solution to the Cauchy problem for the homogeneous heat equation, provided that the initial data are suitably small.

2. 7) = v(^^) I where ^ 5 . 8) 0 < /< 7 + sup 0 < /< r II v( t, •) II i(iRn) + ( 1 X I I 0 1*^2 ^ •) II n, dt) ^ " (K ) Here the supremum is taken on the interval [0, T 1 if T is finite and on [0, +oo) if 38 GLOBAL SOLUTIONS FOR NONLINEAR EVOLUTION EQUATIONS T = + 00 respectively. For brevity we only use the notation [0, T ] in what follows. By definition it is easy to see that, for any v e Xg g j , we have v e L “ (0 , T ; VF'J-n- 3. 9) (1 + f)«/2 V e L “ (0 , T ; VF5-'»- 3. 11) £>* ve ¿2(0,7;//S(|R«)) (|jtl = 2).

A). 55) to prescribe ri (/ = 0,1,.... 53). 56). •/|Dn^^O J l l l w l l A n. 47). 4 is finished. 4, we have II II ^s, r(p/i) < C 5 II WII (R/i) • II W II j> ( 4 . 56) holds. Taking r = 2, q = 2, p = +co. 4, we have IIF(w) II //S(|R«) ^ C5 II w II //S(|R") • II vdl IIF(w) II . 62) and IIF(w) II i(|R/i) < Cj IIwII • Iw I . 63) where C5 is a positive constant (depending on Vq). 4: Suppose that G = G(w) is a sufficiently smooth function of w = (Wj.... 40) holds, then G{w) = 0 ( Iw I“), (a > 1 integer).

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