L_p-regularity theory for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

2023/04/24 by Beom-Seok Han, Han, Beom-Seok, Jaeyun Yi +1 · 1 citation
Economics, Econometrics and Finance · Computer Science · Mathematics · #Stochastic processes and financial applications #Advanced Mathematical Modeling in Engineering #Mathematical Biology Tumor Growth

paper · pdf · doi:10.48550/arxiv.2304.11879

Abstract

We study the existence, uniqueness, and regularity of the solution to the stochastic reaction-diffusion equation (SRDE) with colored noise F: ∂_t u = a^iju_xⁱx^j + bⁱ u_xⁱ + cu - b u^1+β + ξu^1+γ F, (t,x)∈ ℝ₊×ℝ^d; u(0,⋅) = u₀, where a^ij,bⁱ,c, b and ξ are C² or L_∞ bounded random coefficients. Here β>0 denotes the degree of the strong dissipativity and γ>0 represents the degree of stochastic force. Under the reinforced Dalang's condition on F, we show the well-posedness of the SRDE provided γ< (κ(β+1))/(d+2) where κ>0 is the constant related to F. Our result assures that strong dissipativity prevents the solution from blowing up. Moreover, we provide the maximal Hölder regularity of the solution in time and space.

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• On the local well-posedness of randomly forced reaction-diffusion equations with L² initial data and a superlinear reaction term

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