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


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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