Weighted maximal Lq(Lp)-regularity theory for time-fractional diffusion-wave equations with variable coefficients

Abstract

We present a maximal Lq(L p)-regularity theory with Muckenhoupt weights for the equation & part;(alpha)(t)u(t,x)=a(ij)(t,x)u (x)i(x)j(t,x)+f(t,x), t > 0, x is an element of R-d. (0.1)Here, & part;(alpha )(t)is the Caputo fractional derivative of order alpha is an element of(0, 2) and aijare functions of (t,x). Precisely, we show that integral (T)( 0) (integral(d)(R)|(1 - delta)gamma/2(u)xx(t, x)|pw(1)(x)dx) w2(t)dt <= N integral( T)(0) (integral(d)(R) |(1 - delta)gamma/2 f (t, x)|pw1(x)dx w2(t)dt, 0where 1 < p, q < infinity, gamma is an element of R, and w1 and w2 are Muckenhoupt weights. This implies that we prove maximal regularity theory, and sharp regularity of solution according to regularity of f . To prove our main result, we also proved the complex interpolation of weighted Sobolev spaces,[H-p0(gamma 0)(w0), H-p1(gamma 1) (w(1))][theta] =H-p(gamma)(w), where theta is an element of (0, 1), gamma 0, gamma 1 is an element of R, p0, p1 is an element of (1, infinity), wi (i = 0, 1) are arbitrary A(pi )weight, and gamma = (1 - theta)gamma 0 + theta gamma 1, 1 /p 1 - theta = /p0 + theta , w(1)/p = w p1 (1-theta) 0 w/ p0 theta 1 . p1