Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term

We study existence and regularity of distributional solutions for possibly degenerate quasi-linear parabolic problems having a first order term which grows quadratically in the gradient. The model problem we refer to is the following (1){ut−div(α(u)∇u)=β(u)|∇u|2+f(x,t),in Ω×]0,T[;u(x,t)=0,on ∂Ω×]0,T[;u(x,0)=u0(x),in Ω. Here Ω is a bounded open set in RN, T>0. The unknown function u=u(x,t) depends on x∈Ω and t∈]0,T[. The symbol ∇u denotes the gradient of u with respect to x. The real functions α, β are continuous; moreover α is positive, bounded and may vanish at ±∞. As far as the data are concerned, we require the following assumptions: ∫ΩΦ(u0(x))dx<∞ where Φ is a convenient function which …

Elliptic problems involving the 1–Laplacian and a singular lower order term

Elliptic equations having a singular quadratic gradient term and a changing sign datum

In this paper we study a singular elliptic problem whose model is \begin{eqnarray*} - \Delta u= \frac{|\nabla u|^2}{|u|^\theta}+f(x), in \Omega\\ u = 0, on \partial \Omega; \end{eqnarray*} where $\theta\in (0,1)$ and $f \in L^m (\Omega)$, with $m\geq \frac{N}{2}$. We do not assume any sign condition on the lower order term, nor assume the datum $f$ has a constant sign. We carefully define the meaning of solution to this problem giving sense to the gradient term where $u=0$, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum $f$ is less regular or the boundedness of the solutions when the datum $f \in L^m (\Omega)$ with …