# Nonlinear heat equation $u_{t} = \Delta(u^{4})$

Consider the nonlinear heat equation $u_{t} = \Delta(u^{4})$ in $\{x \in \mathbb{R}^{3}: |x| < 1\}$ with $u = 0$ on $\{x \in \mathbb{R}^{3}: |x| = 1\}$. The problem I am working on is to show that all solutions to this PDE tend to zero (pointwise) as $t \rightarrow \infty$.

The equation looks like the porous medium equation so I thought about looking at how the Barenblatt solution is derived, but the issue is that this doesn't cover all solutions, so I'm thinking that there is a general argument to prove the above result.

Let $v = u^4$ and rewrite the PDE in terms of $v$ to get: $$v^{-3/4} v_t = \Delta v.$$ Let $\;v = A(t) B(x,y,z)\;$ and separate variables to get: $$A' = 4 A^{7/4} c \quad \mbox{and} \quad B^{3/4} \Delta B = c \quad \mbox{for a constant } c .$$ The DE in $A$ has the solution $\;A(t)^{3/4} = -\frac{4/3}{4 c t + k}\;$ for a constant $k$. So when $t$ goes to infinity $A$ goes to $0$ and therefore also $v$ and $u$.
• You should format your answer using $\rm\LaTeX$. It's a bit hard to read as is. – Cameron Williams Apr 22 '15 at 3:52