A weighted <i>L_q</i>(<i>L_p</i>)-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

초록

We study the fully degenerate second-order evolution equation u(t )= a(ij)(t)u(xixj )+ b(i)(t)u(xi )+ c(t)u + f, t > 0, x is an element of R-d (0.1) given with the zero initial data. Here a(ij)(t), b(i)(t), c(t) are merely locally integrable functions, and (a(ij)(t))(dxd )is a nonnegative symmetric matrix with the smallest eigenvalue delta(t) >= 0. We show that there is a positive constant N such that integral(T)(0)(integral(Rd)(|u(t, x)| + |u(xx)(t, x)|)(p)dx)(q/p )e(-q integral t0c(s)ds)w(alpha(t))delta(t)dt <= N integral(T)(0)(integral(Rd)|f(t, x)|(p)dx)(q/p )e(-q integral t0c(s)ds )w(alpha(t))(delta(t))(1-q)dt, (0.2) where p, q is an element of (1, infinity), alpha(t) = integral(t)(0)delta(s)ds, and w is Muckenhoupt's weight.

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