The influence of data regularity in the critical exponent for a class of semilinear evolutions equations

2019/10/04 by Marcelo Rempel Ebert, Ebert, Marcelo R., Cleverson Roberto da Luz +3 · 1 citation
Mathematics · #35B33 #35B40 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1910.01823

openalex publication_date 2019/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations % u_tt+(-Δ)^δu_tt+(-Δ)^αu+(-Δ)^θu_t=|u_t|^p, t≥ 0, x∈\Rⁿ, % with p>1, 2θ∈ [0, α] and δ∈ (θ,α]. We show that, under additional regularity (H^α+δ(\Rⁿ)∩ L^m(\Rⁿ) )× (H^2δ(\Rⁿ)∩ L^m(\Rⁿ)) for initial data, with m∈ (1,2], the critical exponent is given by p_c=1+(2mθ)/(n). The nonexistence of global solutions in the subcritical cases is proved, in the case of integers parameters α, δ, θ, by using the test function method (under suitable sign assumptions on the initial data).

Citations

• Asymptotic properties for second-order linear evolution problems with\n fractional laplacian operators

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• Non-Existence of Solutions for a Fourth-Order Structurally Damped Equation Under Nonlinear Memory Effect

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