New sharp lifespan estimates for a semilinear heat equation with zero-mass sign-changing initial data
Abstract
We consider the Cauchy problem ut−Δu=|u|p, (t,x)∈(0,T)×Rn, u(0,x)=eps u0(x), x∈Rn, with small sign-changing initial data having zero total mass. We assume u0∈ L1(Rn)∩ L∞(Rn), |x|u0∈ L1(Rn), ∫Rnu0(x) dx=0, and, for the sharp subcritical upper bounds, that ∫Rnx u0(x) dx≠0. Let Teps denote the maximal lifespan. We identify a new transition exponent pm=1+1/(n+1) inside the Fujita range 1<p≤ pF:=1+2/n, and establish the sharp lifespan estimates Tepsasymp eps−(2(p−1))/(2−(n+1)(p−1)), &1<p<pm, [2mm] eps−2/(n+1) (log1/eps)−2/(n+2), &p=pm, [2mm] eps−p(1/(p−1)−n/2)−1, &pm<p<pF, [2mm] exp (eps−p(p−1)), & p=pF. These estimates reveal a lifespan phenomenon that is different from the classical Lee--Ni law for initial data with positive mass. In the zero-mass setting, the leading linear contribution is dipole-like, while the nonlinear source subsequently generates a positive mass of size O(epsp). Their competition produces the additional threshold pm, the logarithmic correction at p=pm, and, at p=pF, a substantially longer critical lifespan with exponent pF(pF−1) instead of the classical Lee--Ni exponent pF−1. The upper estimates are obtained by backward-Gaussian and critical scale-ODE test-function arguments, whereas the lower estimates follow from a unified L1--L∞ bootstrap preserving the zero-mass cancellation of the linear flow and controlling the mass generated by the nonlinear source. Thus, although pF remains the Fujita critical exponent, zero initial mass creates a new quantitative lifespan regime below and at the critical exponent.
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BibTeX
@misc{torebek2026sharp,
title = {{New sharp lifespan estimates for a semilinear heat equation with zero-mass sign-changing initial data}},
author = {Berikbol T. Torebek},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17318},
eprint = {2609.17318},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17318},
}
RIS
TY - UNPB
TI - New sharp lifespan estimates for a semilinear heat equation with zero-mass sign-changing initial data
AU - Berikbol T. Torebek
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17318
UR - https://arxiv.org/abs/2609.17318
AB - We consider the Cauchy problem u_t−Δu=|u|^p, (t,x)∈(0,T)×R^n, u(0,x)=eps u_0(x), x∈R^n, with small sign-changing initial data having zero total mass. We assume u_0∈ L^1(R^n)∩ L^∞(R^n), |x|u_0∈ L^1(R^n), ∫_R^nu_0(x) dx=0, and, for the sharp subcritical upper bounds, that ∫_R^nx u_0(x) dx≠0. Let T_eps denote the maximal lifespan. We identify a new transition exponent p_m=1+1/(n+1) inside the Fujita range 1<p≤ p_F:=1+2/n, and establish the sharp lifespan estimates T_epsasymp eps^−(2(p−1))/(2−(n+1)(p−1)), &1<p<p_m, [2mm] eps^−2/(n+1) (log1/eps)^−2/(n+2), &p=p_m, [2mm] eps^−p(1/(p−1)−n/2)^−1, &p_m<p<p_F, [2mm] exp (eps^−p(p−1)), & p=p_F. These estimates reveal a lifespan phenomenon that is different from the classical Lee--Ni law for initial data with positive mass. In the zero-mass setting, the leading linear contribution is dipole-like, while the nonlinear source subsequently generates a positive mass of size O(eps^p). Their competition produces the additional threshold p_m, the logarithmic correction at p=p_m, and, at p=p_F, a substantially longer critical lifespan with exponent p_F(p_F−1) instead of the classical Lee--Ni exponent p_F−1. The upper estimates are obtained by backward-Gaussian and critical scale-ODE test-function arguments, whereas the lower estimates follow from a unified L^1--L^∞ bootstrap preserving the zero-mass cancellation of the linear flow and controlling the mass generated by the nonlinear source. Thus, although p_F remains the Fujita critical exponent, zero initial mass creates a new quantitative lifespan regime below and at the critical exponent.
ER -