Linear and Quasi-linear Equations of Parabolic Type

a₁ analogous arbitrary assume assumptions belongs boundary conditions boundary value problems bounded bounded functions boundedness c₁ Cauchy problem Chapter coefficients and free consider continuous function converge cylinder Q d2Zo differential domain dx dt dxdt elements equal to zero equation 0.1 estimate of max finite follows free terms fulfilled func function u(x heat equation Hölder condition Hölder continuous Hölder's inequality identity inequality integral K₁ L₂(QT Lemma linear equations M₁ max u(x norm obtain operator parabolic equations parameters positive number proof proved quantities quasi-linear equations respect right side satisfy conditions satisfying the conditions smooth function solution of problem solution u(x space t₁ Theorem 6.1 tions unique solvability valid virtue vrai max αλ ατ Κρ μ₁ μι ди дх дхі