Abstract
In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: {ut=αuxx+β[φ(u)]xx+f(u)inQ:=Ω×(0,T),u=0on∂Ω×(0,T),u(x,0)=u0(x)inΩ,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u_{t}=\\alpha u_{xx}+\\beta [\\varphi (u) ]_{xx}+f(u) &\\text{in} \\ Q:=\\Omega \\times (0,T), \\\\ u=0 &\\text{on} \\ \\partial \\Omega \\times (0,T), \\\\ u(x,0)=u_{0}(x) &\\text{in} \\ \\Omega , \\end{cases} $$\\end{document} where T>0, Omega subset mathbb{R} is a bounded interval, u_{0} is nonnegative bounded Radon measure on Ω, and alpha , beta geq 0, under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.
Highlights
In this paper we address the existence, decay estimate, and the asymptotic behavior of solutions for the following problem:⎧ ⎪⎪⎨ut = [ψ(u)]xx + f (u) in Q := × (0, T),⎪⎪⎩uu(=x,00) = u0(x) on ∂ × (0, T), in, (P)where T > 0, ⊂ R is a bounded interval, u0 is nonnegative bounded Radon measure on under suitable assumptions on ψ and f expressed as follows: ψ(s) = αs + βφ(s) (1.1)Nkombo et al Advances in Difference Equations (2021) 2021:509 for any (α, β) ∈ R+ × R+, where the function φ satisfies the following assumptions: ⎧ (H) ⎪⎪⎪⎪⎪⎨((iii))φ ∈ L∞(R+) ∩ C2(R+), φ, φ ∈ L∞(R+), φ(0) = 0
In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above
1 Introduction In this paper we address the existence, decay estimate, and the asymptotic behavior of solutions for the following problem:
Summary
Throughout this paper, we consider the solutions of problem (P) as maps from (0, T) to the cone of nonnegative bounded Radon measure on , which verify (P) in the following sense: for a suitable class of test function ξ , there holds. In [1], the authors proved the existence and qualitative properties of the Radon measure-valued solutions associated with the Young measure Another difference between problem (A.4) and (P) is the assumption which fulfills the function F(x, t, u). In [17] the authors showed the existence, qualitative properties, and decay estimate of the Radon measure-valued solutions to the Cauchy problem of (A.5).
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