Sharp behavior of semilinear damped wave equations driven by mixed local-nonlocal operators

2025/09/29 by Wenhui Chen, Chen, Wenhui, Tuan Anh Dao +1
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2509.24940

Abstract

This paper investigates the Cauchy problem for the semilinear damped wave equation u_tt+L_a,bu+u_t=|u|^p with the mixed local-nonlocal operator L_a,b:=-aΔ+b(-Δ)^σ, where a,b∈ℝ₊ and σ∈(0,1)∪ (1,+∞). We determine the critical exponent for this problem being p_crit=1+\frac2min\1,σ\n, which sharply separates global in-time existence and finite-time blow-up of solutions. Furthermore, for the super-critical case p>p_crit, we establish the asymptotic profiles of global in-time solutions, showing the anomalous diffusion when σ∈(0,1) and the classical diffusion when σ∈(1,+∞), together with the sharp decay estimates. For solutions blowing up in finite time when 1

Citations

• Sharp lifespan estimates for semilinear fractional evolution equations with critical nonlinearity
• Fujita exponent and blow-up rate for a mixed local and nonlocal heat equation
• Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators
• Description of an ecological niche for a mixed local/nonlocal dispersal:\n an evolution equation and a new Neumann condition arising from the\n superposition of Brownian and L 'evy processes
• A Faber-Krahn inequality for mixed local and nonlocal operators
• (Non)local logistic equations with Neumann conditions
• A test function method for evolution equations with fractional powers of the Laplace operator
• Mixed local and nonlocal elliptic operators: regularity and maximum\n principles
• A blow-up result for semi-linear structurally damped σ-evolution equations
• The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in high dimensions

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