An example of Kirchhoff equation

Question

I'm studying about a simply kind of Kirchhoff equation in one-dimensional that means $$\begin{cases} u_{tt}(x,t)-\left(1+\int_{0}^{1}{u^2_x(x,t)dx}\right)u_{xx}(x,t)=|u(x,t)|^{p-2}u(x,t),0<x<1,0<t<T\\ u_x(0,t)=u(1,t)=0\\ u(x,0)=f(x),u_t(x,0)=g(x),\end{cases}$$ where $p\ge 3,u\in V\equiv \left\{v\in H^1:v(1)=0\right\}$. In details, I'm finding an example of this equation, but I'm being in stalemate with the case of $p=3$. So can anybody help me?

Answer 1

Partial answer: This is a nonlinear hyperbolic equation modeling traversal vibrations/oscillations. I highly doubt there exists an explicit solution for this type of nonlinear equation (I might be wrong). Many treatises dealing with a general version of your equation: $$ \frac{\partial^2 u}{\partial t^2} - M(t,\int_{\Omega}|\nabla u(x,t)|^2\,dx)\Delta u = f(x,u) $$ point to the reference of Jacques-Louis Lions's really old article On some questions in boundary value problems of mathematical physics in Contemporary developments in continuum mechanics and partial differential equations. I just checked the math library here in my university, this article didn't contain an explicit formula either. Also most papers only have qualitative estimates for this type of equation. For the stationary problem, letting $t\to \infty$, and $u$ doesn't change w.r.t. time anymore so the time derivative term is gone: $$ -M(\int_{\Omega}|\nabla u(x)|^2\,dx)\Delta u = |u|^{p-1}u, $$ for example this paper studies a semilinear problem related to above equation, when $u$ is positive (so you can get rid of the absolute value). A related equation can be further simplified if we can prescribe $\|u\|_{L^2}$ a priori: $$ -u_{xx} = |u|^{p-2} u, $$ the positive solution to this when $p=3$ is the Weierstrass Elliptic Function, the bad thing for nonlinear wave equation is that we can't construct the solution like plane wave in the linear wave equation, a.k.a. $u(x,t) = \mathfrak{R}\big(e^{-i\omega t} \phi(x)\big)$.

My suggestion: if you wanna examples (I am assuming you are looking what the solution looks like), search for the literatures that apply some numerics modeling this kind of the nonlinear elastic wave.