## Positivity versus localization in degenerate diffusion equations.(English)Zbl 0596.35073

The authors study the Cauchy problem (CP) $$u_ t-\Delta (u^ m)+f(u)=0$$ in $$S=R^ N\times (0,\infty)$$, $$u(x,0)=u_ 0(x)$$ for $$x\in R^ N$$, where $$m\geq 1$$, $$N\geq 1$$, $$f\in C^ 1([0,\infty))$$, $$f(0)=0$$ and $$u_ 0$$ is a nonnegative continuous function on $$R^ N$$ with bounded (and nonempty) support. This problem has the positivity property (P) if (for each $$u_ 0$$ as above) for each $$x\in R^ N$$ there is T(x)$$\geq 0$$ such that $$u(x,t)>0$$ for all $$t\geq T(x)$$. It is known that (P) holds in the following cases: 1) $$m=1$$ (then T(x)$$\equiv 0)$$; 2) $$f\leq 0$$; $$3)\quad \limsup_{s\searrow 0}s^{-m} f(s)<\infty.$$
The problem (CP) has the localization property (L) if (for each $$u_ 0)$$ the solution u has the property that there is $$R>0$$ such that $$\sup p u(.,t)\subset B_ R=\{x\in R^ N:\quad | x| \leq R\}$$ for all $$t\geq 0$$. Let $$f(s)=s^ m g(s)$$ $$(s>0)$$ where $$g: R^+\to R^+$$ satisfies for some $$s_ 0>0:$$ (a) g’(s)$$\leq 0$$ when $$0<s<s_ 0$$; (b) $$s^{\alpha} g(s)$$ is nondecreasing on $$(0,s_ 0)$$ for some $$\alpha\in (0,m-1)$$. Then
(1) (L) holds iff $$\int_{0}ds/F(s)^{1/2}<\infty$$, where $$F(s)=\int^{s}_{0}f(r^{1/m})dr;$$
(2) there are constants $$0<A<B<\infty$$ such that $$A\omega$$ (t)$$\leq \leq R^-(t)\leq R^+(t)\leq B\omega (t)$$ for $$t\geq 1$$, if $$\int_{0}ds/F(s)^{1/2}=\infty$$, where $$\omega (t)=\Phi (y(t,1),1),\quad \Phi (s,v)=\int^{v}_{s}dr/(rg(r)^{1/2}),\quad R^-(t)=\sup \{R>0:\quad B_ R\subset P(t)\},\quad R^+(t)=\inf \{R>0:\quad P(t)\subset B_ R\}$$ and $$P(t)=\{x\in R^ N:\quad u(x,t)>0\}.$$ Here y(t,1) is the solution of $$y'=-f(y)$$, $$y(0)=1$$.
Reviewer: J.Danesova

### MSC:

 35K65 Degenerate parabolic equations 76R99 Diffusion and convection 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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