New explicit solutions of quasilinear heat equations with general first-order sign-invariants

Abstract

For the one-dimensional quasilinear heat equations u t = ψ( u) u xx + f( u), with smooth functions ψ( u) ≥ 0 and f( u), we derive conditions under which a general quadratic Hamilton-Jacobi operator H( u) ≡ u t − [ h 2( u)( u x ) 2 + h 1( u) u x + h 0( u)] (with the functions h 2, h 1, h 0 to be determined) preserves both the signs, ≥ 0 and ≤ 0, on the solutions u( x, t). H( u) is then called a sign-invariant of the given parabolic equation. Several types of different sign-invariants are established which are governed by a linear second-order ordinary differential equations. New explicit solutions via finite-dimensional dynamical systems corresponding to zero-preserving of the first-order operator are constructed.