Linear and Quasi-linear Equations of Parabolic Type

analogous arbitrary assume assumptions Banach space belongs boundary conditions boundary value problems bounded bounded function boundedness Cauchy problem Chapter classical solution coefficients and free conditions of Theorem consider continuous function converge cylinder Q defined denote derivatives determined domain Q dx dt dxdt elements elliptic equal to zero equation 1.1 estimate of max example finite formulate free terms fulfilled func function u(x fundamental solution heat equation Holder condition identity initial condition integral known functions Lemma linear equations matrix maximum principle maxQ method nonnegative norm obtain operator parabolic equations parabolic systems parameters polynomial positive number properties proved quantities quasi-linear equations respect result right side satisfy conditions satisfy the conditions smooth function solution of problem solution u(x space subdomain Suppose tend to zero Theorem 6.1 tion transform unique solvability valid vector virtue vrai max x)dy