The regularity of generalized solutions for the n-dimensional quasi-linear parabolic diffraction problems

In this paper, we study the regularity of generalized solutions u(x,t) for the n-dimensional quasi-linear parabolic diffraction problem. By using various estimates and Steklov average methods, we prove that (1): for almost all t the first derivatives ux(x,t) are Hölder continuous with respect to x up to the inner boundary, on which the coefficients of the equation are allowed to be discontinuous; and (2): the first derivative ut(x,t) is Hölder continuous with respect to (x,t) across the inner boundary.