In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problemwhere N ≥ 1, 0 < θ ≤ 2, p > 1 and µ is a Radon measure or a measurable function in R N . Our conditions lead optimal estimates of the life span of the solution with µ behaving like λ|x| −A (A > 0) at the space infinity, as λ → +0.

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We investigate some geometric properties of level sets of the solutions of parabolic problems in convex rings. We introduce the notion of parabolic quasi-concavity, which involves time and space jointly and is a stronger property than the spatial quasi-concavity, and study the convexity of superlevel sets of the solutions.

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We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation,whereAssume that u is a solution of (P) satisfyingfor some constants C > 0 and A > 1. Then it is well known that the solution u behaves like the heat kernel. In this paper we give the ([K ] + 2)th order asymptotic expansion of the solution u, and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term F. Here [K ] is the integer satisfying K − 1 < [K ] ≤ K .

This paper is concerned with a nonlinear integral equation) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P ) behaving like a multiple of the integral kernel G as t → ∞.

In this paper we consider degenerate parabolic equations, and obtain an interior and a boundary Harnack inequalities for nonnegative solutions to the degenerate parabolic equations. Furthermore we obtain boundedness and continuity of the solutions.

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