The nonnegative weak solution of a degenerate parabolic equation with variable exponent growth order

Abstract

AbstractA degenerate parabolic equation of the form $$\bigl( \vert v \vert ^{\beta-1}v\bigr)_{t}= \operatorname{div} \bigl(b(x,t) \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)+\nabla\vec{g}\cdot\nabla\vec{\gamma}(v) $$(|v|β−1v)t=div(b(x,t)|∇v|p(x,t)−2∇v)+∇g→⋅∇γ→(v) is considered, where $\vec{g}=\{g^{i}(x,t)\}$g→={gi(x,t)}, $\vec{\gamma}(v)=\{ \gamma_{i}(v)\}$γ→(v)={γi(v)}. If the diffusion coefficient $b(x,t)\geq0$b(x,t)≥0 is degenerate on the boundary, by adding some restrictions on $b(x,t)$b(x,t) and g⃗, the existence and uniqueness of weak solutions are proved. Based on the uniqueness, the stability of weak solutions can be proved without any boundary condition.