Initial traces and solvability for a semilinear heat equation on a half space of ℝ^{ℕ}

Abstract

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.

Optimal singularities of initial data for solvability of the Hardy parabolic equation

“…Compare with [11]. In order to prove Theorem 3.1, we first modify the arguments in [11] and prove Proposition 3.1 below. Proposition 3.1.…”

“…Of course, in the case of Ω = R N , we ignore the boundary condition, and in the case where θ is a positive even integer, R N \ Ω in the boundary condition is replaced by ∂Ω. Among them, the author of this paper, Ishige, and Takahashi [11] considered the solvability of the Cauchy-Dirichlet problem for…”

“…For this reason, we could not treat initial data of (1.2) in the framework of the Radon measure on Ω. In order to solve this problem the authors of [11] introduced the following idea. By the explicit formula of p(x, y, t) we see that for y = (y ′ , 0) ∈ ∂Ω,…”

Solvability of superlinear fractional parabolic equations

“…Results of this type are important for the development of any general function-analytic framework for well-posedness of problem (SHE), for example in identifying an optimal Orlicz class of initial data for a given nonlinearity f . Such studies exist for a variety of semilinear problems: see [6,8] for problem (SHE) with power law and power-log nonlinearities; see [10,11] for semilinear parabolic systems; [15] for the linear heat equation with nonlinear boundary conditions; [17,18] for the Hardy parabolic equation; [22] for higher-order semilinear parabolic equations; [16] for semilinear heat equations in a half-space of R N .…”

“…Alternatively, setting µ c = λ 0 µ 0 we see that problem (P) with µ = λµ c is locally solvable for all λ ∈ (0, 1) and not locally solvable for any λ > 1. This type of parameterisation, along the ray {λµ c } λ>0 , was considered in [16] for problem (P) on a half-space, where µ c is referred to as an 'optimal singularity'. See also [6,18].…”