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.
In this paper we obtain necessary conditions and sufficient conditions for the solvability of the problemwhere N ≥ 1, p > 1 and µ is a nonnegative measurable function in R N + or a Radon measure in R N with supp µ ⊂ D. Our sufficient conditions and necessary conditions enable us to identify the strongest singularity of the initial data for the solvability for problem (P). Furthermore, as an application, we obtain optimal estimates of the life span of the minimal solution of (P) with µ = κϕ as κ → 0 or κ → ∞.
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of
R
N
\mathbb {R}^N
under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy–Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy–Dirichlet problem.
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.
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