## 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

### Keywords:

Cauchy problem; maximum principle; parabolic inequality

### Citations:

Zbl 0267.35053; Zbl 0252.35005
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