(Crossposted from http://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$.

Consider the PDE $$u_t = \Delta F(u) \quad \text{on $\Omega$}$$ $$u(x,0) = u_0$$ $$u(x,t) = C_1\quad \text{on $\partial \Omega$}$$

Suppose we have two weak solutions $u$ and $v$ in $L^2(0,T;H^1)\cap H^{-1}(0,T;H^{-1})$ with initial data $u_0$ and $v_0$ and boundary data $C_u$ and $C_v$ respectively, where $u_0 \leq v_0$ and $C_u \leq C_v$.

How do I show that the comparison principle holds: that $u \leq v$?

Assume more smoothness of the solutions if necessary. I can't do it. I know we need to test with $(u(t)-v(t))^+$ but I don't know what to do with the nonlinear term.

(Crossposted from https://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$.

Consider the PDE $$u_t = \Delta F(u) \quad \text{on $\Omega$}$$ $$u(x,0) = u_0$$ $$u(x,t) = C_1\quad \text{on $\partial \Omega$}$$

Suppose we have two weak solutions $u$ and $v$ in $L^2(0,T;H^1)\cap H^{-1}(0,T;H^{-1})$ with initial data $u_0$ and $v_0$ and boundary data $C_u$ and $C_v$ respectively, where $u_0 \leq v_0$ and $C_u \leq C_v$.

How do I show that the comparison principle holds: that $u \leq v$?

Assume more smoothness of the solutions if necessary. I can't do it. I know we need to test with $(u(t)-v(t))^+$ but I don't know what to do with the nonlinear term.

Let $\Omega$ be a bounded smooth domain and let $p < 0$ be real. Suppose that $u, v \in L^2(0,T;H^1(\Omega) \cap L^2(0,T;L^2(\partial\Omega))$ with $|u|^{p+1} \in L^2(0,T;L^2(\partial\Omega))$ and likewise for $v$.

Suppose $u(0) \leq v(0)$ a.e and $$-\int_0^T \int_{\partial\Omega} (|u|^pu - |v|^pv)\varphi' + \int_0^T \int_{\Omega}\nabla (u-v)\nabla \varphi \leq \int_{\partial\Omega}(|u(0)|^{p+1}-|v(0)|^{p+1})\varphi(0)$$ for all $\varphi \in C^1(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$ with $\varphi \geq 0$ and $\varphi(T) = 0$.

How do I show that $u \leq v$ a.e?

It seems like one should test with $\varphi = (|u|^pu - |v|^pv)^+$ however this causes problems with the second term on the left..

(Crossposted from http://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$.

Consider the PDE $$u_t = \Delta F(u) \quad \text{on $\Omega$}$$ $$u(x,0) = u_0$$ $$u(x,t) = C_1\quad \text{on $\partial \Omega$}$$

Suppose we have two weak solutions $u$ and $v$ in $L^2(0,T;H^1)\cap H^{-1}(0,T;H^{-1})$ with initial data $u_0$ and $v_0$ and boundary data $C_u$ and $C_v$ respectively, where $u_0 \leq v_0$ and $C_u \leq C_v$.

How do I show that the comparison principle holds: that $u \leq v$?

Assume more smoothness of the solutions if necessary. I can't do it. I know we need to test with $(u(t)-v(t))^+$ but I don't know what to do with the nonlinear term.