Maximal Sign-Invariants and New Exact Solutions of Quasilinear Parabolic Equations with Gradient Diffusivity

We consider quasilinear parabolic equations with gradient-like diffusivity u_t=div(|∇_u|^σ ∇_u)+f(u),   x ∋ <b>R</b>^N, t > 0, when σ ≠-1 is a fixed constant and f(u) is a given smooth function. We also study quasilinear parabolic equations with a gradient-dependent coefficient u_t = h(|∇_u|Δu + f(u), with a smooth function h(p). For both classes of equations we derive first-order sign-invariants, i.e. first-order operators preserving their signs on the evolution orbits {u(•, t), t>0}. We give a complete description of (maximal) sign-invariants of prescribed structures. As a consequence, we construct new exact solutions on some quasilinear equations.