I know that for the following pde $u_{t}=(u^{\alpha})_{xx}$ where $\alpha \geqslant 0$ is not an integer, one can seek for solutions of the form $u(t,x)=t^{a}(1-x^2 t^{b})_{+}^{c}$ so that it produces non trivial solutions of this pde.
I am wondering if someone know how to find non trivial solutions of the following pde : $$u_{tt}=(u^{\alpha})_{xx}$$
It seems very similar to the previous, but I tried a lot of solution profiles and none of them worked.
Is there an automatic way of producing non trivial solution to this pde ? If you know so feel free to write it !
Thanks.