Critical non-linearity for some evolution equations with Fujita-type critical exponent

2024/04/09 by Girardi, Giovanni · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.06458

Abstract

We consider the Cauchy problem for a class of non-linear evolution equations in the form L(∂_t,∂_x) u=F(∂_t^ℓ u), (t,x)∈ [0,∞)× ℝⁿ; here, L(∂_t,∂_x) is a linear partial differential operator with constant coefficients, of order m≥ 1 with respect to the time variable t, and ℓ is a natural number satisfying 0≤ ℓ≤ m-1. For several different choices of L, many authors have investigated the existence of global (in time) solutions to this problem when F(s)=|s|^p is a power non-linearity, looking for a critical exponent p_c>1 such that global small data solutions exist in the supercritical case p>p_c, whereas no global weak solutions exist, under suitable sign assumptions on the data, in the subcritical case 1

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