We prove that the thin film equation ht+div (hn grad (Δh))=0 in dimension d[ges ]2 has a unique C1 source-type radial self-similar non-negative solution if 0<n<3 and has no solution of this type if n[ges ]3. When 0<n3 the solution h has finite speed of propagation and we obtain the first order asymptotic behaviour of h at the interface or free boundary separating the regions where h>0 and h=0. (The case d=1 was already known [1]).

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We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: ut = J * u − u , where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) giving explicit unbounded polynomial solutions.

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Abstract. In this paper we study the blow-up phenomenon for nonnegative solutions to the following parabolic problem:After discussing existence and uniqueness we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p + > 1.When Ω = R N we show that if p− > 1 + 2/N then there are global nontrivial solutions while if 1 < p − ≤ p + ≤ 1 + 2/N then all solutions to the problem blow up in finite time. Moreover, in case p− < 1+2/N < p+ there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global nontrivial solutions.When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough then the problem possesses global nontrivial solutions regardless the size of p(x).