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On some properties of solutions of the Cauchy problem for a quasilinear parabolic equation. (English) Zbl 0565.35048

The author considers the Cauchy problem \[ u_ t=\sum^{n}_{i,j=1}a_{ij}(t,x,u,u_ x)u_{x_ ix_ j}+b(t,x,u,u_ x)+B(t,u)\quad in\quad P:={\mathbb{R}}^ n\times (0,T] \] in the class of functions u satisfying \(| u(x,t)| \leq D \exp (d| x|^ 2).\) Using the maximum principle, he proves, among others: (a) If u(0,x) is monotone \((n=1)\), then u(t,x) is monotone in the same manner (t fixed); (b) if \(m_ 1\leq u(0,x)\leq m_ 2\) and if \(v_ 1\), \(v_ 2\) are solutions of \(v'=B(t,v)\) satisfying \(v_ i(0)=m_ i\), then \(v_ 1(t)\leq u(t,x)\leq v_ 2(t);\) (c) if u(0,x) is Lipschitz continuous and \(B=B(t)\), then \(| u_ x(x,t)| \leq C.\)
Remark. Closely related results are well known. For (a), see R. M. Redheffer and W. Walter, Math. Ann. 209, 57-67 (1974; Zbl 0267.35053), in particular Corollary 3 and p. 66. (b) follows immediately from Nagumo’s lemma; see Lemma 28.XV in W. Walter, Differential and integral inequalities (1970; Zbl 0252.35005). The simple method used by the author - considering the function \(w(x,y,t)=u(x,t)-u(y,t)\) and deriving a parabolic inequality for w - deserves to be more widely known.
Reviewer: W.Walter

MSC:

35K15 Initial value problems for second-order parabolic equations
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35B50 Maximum principles in context of PDEs
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