Solvability of a Semilinear Parabolic Equation with Measures as Initial Data
We study a sharp condition for the solvability of the Cauchy problem \(u_t-\varDelta u=u^p\) , \(u(\cdot ,0)=\mu \) , where \(N\ge 1\) , \(p\ge (N+2)/N\) and \(\mu \) is a Radon measure on \(\mathbf {R}^N\) . Our results show that the problem does not adm
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Abstract We study a sharp condition for the solvability of the Cauchy problem ut − Δu = up , u(·, 0) = μ, where N ≥ 1, p ≥ (N + 2)/N and μ is a Radon measure on RN . Our results show that the problem does not admit any local nonnegative solutions for some μ satisfying μ({y ∈ RN ; |x − y| < ρ}) ≤ Cρ N−2/(p−1) (log(e + 1/ρ))−1/(p−1) (x ∈ RN , ρ > 0) with a constant C > 0. On the other hand, the problem admits a local solution if μ({y ∈ RN ; |x − y| < ρ}) ≤ Cρ N−2/(p−1) (log(e + 1/ρ))−1/(p−1)−ε (x ∈ RN , ρ > 0) with a constant ε ∈ (0, 1/(p − 1)). Keywords Solvability · Semilinear parabolic equations · Radon measures 2010 MSC: 35K58 · 35K15 · 35A01
1 Introduction This paper concerns the Cauchy problem
ut − Δu = up in RN × (0, T ), u(·, 0) = μ in RN ,
(1)
where N ≥ 1, T > 0, p > 1 and μ is a nonnegative Radon measure on RN . We say that u is a solution of (1) if u is a nonnegative function satisfying the equation in the classical sense and u(·, t) → μ as t ↓ 0 weakly as measures on each fixed open ball. In this paper, we study a sharp condition on μ for the solvability of (1) under the condition that p ≥ pF := (N + 2)/N.
J. Takahashi (B) Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan e-mail: [email protected] © Springer International Publishing Switzerland 2016 F. Gazzola et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer Proceedings in Mathematics & Statistics 176, DOI 10.1007/978-3-319-41538-3_15
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J. Takahashi
A necessary and sufficient condition on μ for the existence of solutions of (1) was indirectly characterized by Baras and Pierre [4, THÉORÈME 3.2]. They also gave an explicit necessary condition for existence in [4, PROPOSITION 3.2] (see also Andreucci and DiBenedetto [3, Part I, Proposition 4.3, Remark 4.9]). More precisely, they proved that if u ≥ 0 satisfies ut − Δu = up in RN × (0, T ), then there exists a unique Radon measure μ on RN such that u(·, t) → μ as t ↓ 0 weakly as measures on each fixed open ball. Furthermore, μ must satisfy sup μ(B1 (x)) < +∞
if p < pF ,
(2)
x∈RN
and for any compact subset K of RN there exists a constant C > 0 such that μ(Bρ (x)) ≤
C(log(1/ρ))− 2
N
Cρ
2 N− p−1
if p = pF , if p > pF ,
(3)
for any x ∈ K and ρ > 0 small. Here Bρ (x) is the N-dimensional open ball of radius ρ > 0 centered at x ∈ RN . We remark that (2) is also a sufficient condition for the existence of local solutions of (1) (see [4, COROLLAIRE 3.4.i]). For p > pF , by the result of Robinson and Sier˙ze˛ga [17, Theorem 3], we can check that the problem (1) admits a solution for the initial data μ1 (A) := A |f1 (x)|dx if |f1 (x)| ≤ c|x|−2/(p−1) with c > 0 small (see Proposition 2 in the last part of Sect. 3 for more details). Therefore it is expected that (3) is also a sufficient condition. In this paper, by taking an approach similar to that of Kan and the author [11], we first show that (3) is not a sufficient condition. Indeed, Theorem 1 below says that the problem (1) does not admi
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