Estimates for Solutions to Partial Quasilinear Differentila Equations

Abstract

Let’s consider L(u) = a(x, t)utt + b(x, t)utx + c(x, t)uxx + d(x, t)ut + h(x, t)ux, x ∈ R1. The second order quasilinear equations are studied in the following form: L(u) + r(u) a(x, t)u 2 t + b(x, t)utux + c(x, t)u 2 x + f(x, t, u) = 0, x ∈ R1 (1). The objective is to reduce this equation to a linear equation and to study the solutions of the equation (1) depending on the solutions of the linear equation obtained and the functions r(u) and f(x, t, u). For this purpose we make the substitution u = z(v), v = v(x, y) (2).