Cauchy–Dirichlet Problem to Semilinear Multi-Term Fractional Differential Equations

Fractal and Fractional

article

Nataliya Vasylyeva¹ 

[1] Institute of Applied Mathematics and Mechanics of NASU;Dipartimento di Matematica

关键词: a priori estimates;    Caputo derivatives;    nonlinear oxygen subdiffusion;    global classical solvabilityPrimary 35R11;    35B45;    Secondary 35B655;    26A33;    35Q92;   

DOI  :  10.3390/fractalfract7030249

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain Ω ⊂R n  in the unknown u = u ( x , t ),  Dt ν 0(ϱ 0u )−Dt ν 1(ϱ 1u )−L 1u −∫0tK( t − s ) L 2u( x , s )d s = f( x , t )+ g ( u ),0 <ν 1<ν 0< 1 , where Dt ν i  are the Caputo fractional derivatives,  ϱ i=ϱ i ( x , t )  with  ϱ 0≥μ 0 0, and L i are uniform elliptic operators with time-dependent smooth coefficients. The principal feature of this equation is related to the integro-differential operator  Dt ν 0(ϱ 0u )−Dt ν 1(ϱ 1u ) , which (under certain assumption on the coefficients) can be rewritten in the form of a generalized fractional derivative with a non-positive kernel. A particular case of this equation describes oxygen delivery through capillaries to tissue. First, under proper requirements on the given data in the linear model and certain relations between ν 0 and ν 1, we derive a priori estimates of a solution in Sobolev–Slobodeckii spaces that gives rise to providing the Hölder regularity of the solution. Exploiting these estimates and constructing appropriate approximate solutions, we prove the global strong solvability to the corresponding linear initial-boundary value problem. Finally, obtaining a priori estimates in the fractional Hölder classes and assuming additional conditions on the coefficients ϱ 0 and ϱ 1 and the nonlinearity g ( u ), the global one-valued classical solvability to the nonlinear model is claimed with the continuation argument method.

【 授权许可】

CC BY