I have the following two PDEs which I want to check for stability:$$u_t= u_{xx} , \ u(x,0)=1 , \ u_x(1,t)=-hu^4(1,t) , \ u_x(0,t) = 0 $$$$u_t = u_{xx}-\sin(x+t)+\cos(x+t) , \ u(x,0)=\cos(x) ,\ u_x(1,t)=-hu^4(1,t)-\sin(1+t)+h\cos^4(1+t) , \ u_x(0,t)=-\sin(t)$$For the first I got that $d/dt (u,u) = d/dt \int_0^1 u^2 dx = 2[-hu^5(1,t)-\int_0^1 u_x^2dx$, but how to estimate $u^5(1,t)$?Just before I forget, h is some parameter greater than zero.Thanks!
Stability of two PDEs
pde;parabolic pde