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On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources. (English) Zbl 0762.35052

The paper concerns the Cauchy problem \[ (1)\quad u_ t-\Delta u^ m=u^ p/(1+| x|)^ \alpha\quad\text{in }\mathbb{R}^ N\times(0,T),\qquad (2)\quad u(0)=u_ 0\quad\text{in }\mathbb{R}^ N \] \((m\geq 1,\;p>1\), \(\alpha\) any real number).
The authors prove the existence local in time of a weak solution of (1), (2) for initial values \(u_ 0\in L^ q_{loc}(\mathbb{R}^ N)\) \((q\geq 1)\) or \(u_ 0=\mu\) a \(\sigma\)-finite Borel measure in \(\mathbb{R}^ N\), where the integral average of \(u_ 0\) over a ball \(B_ r(x)\) has to satisfy an additional condition when \(| x|\to\infty\) (for instance, if \(\alpha<0\) then \(u_ 0\) has to decay nearly pointwise as fast as \((1+| x|)^{\alpha/(p-1)}\) when \(| x|\to\infty\); if \(0<\alpha<2(p-1)/(m-1)\), \(m>1\), then it is sufficient that \(u_ 0\) is locally bounded and grows not faster than \((1+| x|)^{\alpha/(p- 1)}\) as \(| x|\to\infty\).
Then two results on existence and non-existence of global solutions of (1), (2) are presented \((m<p<m+((2-\alpha)/N)\), and \(\alpha=0\) with \(1<p<m+(2/N)\) or \(p>m+(2/N))\).
A second group of results concerns the behaviour at infinity of non- negative supersolutions of (1), \(L^ \infty_{loc}\)-estimates and behaviour near \(t=0\) of non-negative subsolutions of (1) and the existence of an initial trace (in the sense of measures) of any non- negative weak solution of (1). A uniqueness result for weak solutions is established under some additional conditions pointwise in \(t\) on the solution. Most of the existence results continue to hold for more general parabolic equations in divergence form with the above type of degeneration.
Reviewer: J.Naumann (Berlin)

MSC:

35K65 Degenerate parabolic equations
35K15 Initial value problems for second-order parabolic equations
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