Heat Equation With a Nonlinear Term

Let $\Omega \subset \mathbb{R}^n$ bounded domain with smooth boundary. Show that there is at most one solution to the problem

\begin{equation} \left\{ \begin{aligned} &u_{t} - \Delta u + cos(u)= 0, \ \ \ \ \Omega \times (0, T),\\ & u(x,t) = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \partial \Omega \times (0, T) \\ & u(x,0) = u_0(x) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Omega. \end{aligned} \right. \end{equation}

In this answer Uniqueness of a nonlinear heat equation?, why does the integral $\int v \Delta v dx$ vanish?

Solution 1:

I don't think it's so much that this integral vanishes, but rather that it's negative. This is an application of Green's identity:

$$\begin{align} \int_\Omega v\cdot\Delta v\,dx & = -\int_\Omega |\nabla v|^2dx + \int_{\partial\Omega}v\,\nabla v\cdot n\,dx \\ & = -\int_\Omega|\nabla v|^2dx \\ \end{align}$$

The boundary integral vanishes because we assumed that $v = u_1 - u_2$ is the difference of two hypothetical solutions satisfying the same Dirichlet boundary conditions. So to fill out the chain of inequalities from the original post just a bit:

$$\begin{align} \frac{1}{2}\frac{d}{dt}\int_\Omega v^2dx & = \int_\Omega v\cdot\partial_tv\, dx \\ & = \int_\Omega v\cdot(\partial_t v - \Delta v + \Delta v)dx \\ & \le \int_\Omega v\cdot\Delta v\,dx + \int_\Omega|v||\partial_t v - \Delta v|dx \\ & = -\int_\Omega|\nabla v|^2dx + \int_\Omega|v||\partial_t v - \Delta v|dx \\ & \le \int_\Omega|v||\partial_tv - \Delta v|dx \end{align}$$

and from there you apply the rest of the specific knowledge from the problem, like the fact that the nonlinear term is cosine which is a Lipschitz function.

To use a little more fancy terminology, the Laplace operator acting on function spaces where any of the usual boundary conditions apply (Dirichlet, Neumann, or Robin) is symmetric and negative-definite. This is also true of other divergence-form elliptic operators; it's tremendously useful for proving stability properties about the solution and for numerical analysis.