# Stability of two PDEs

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!

• Probably better to ask this on math.SE – David Ketcheson Mar 11 '16 at 5:16
• What do you mean by stability? Smooth dependence of the solutions on the data? Or that the solutions do not grow arbitrarily? – Jan Mar 11 '16 at 14:40
• @Jan by stability I mean the problem is well posed; i.e it has a unique solution and for the first PDE we have the estimate $\| u(\cdot , t)\| \le Ke^{\alpha t}\| u(\cdot , 0) \|$; for the second PDE with Duhamel term, the estimate is different; it should be for $u_t = P(x,t,\partial_x)u + F(x,t)$ for the problem to be well posed it should have unique solution and satisfy: $\| u(\cdot , t) \| \le Ke^{\alpha (t-t_0)}(\| u(\cdot , t_0) \| +\max_{t_0\le \tau \le t} \| F(\cdot , \tau)\|$; I am using the terminology from here:tinyurl.com/h57zpgl on pages 142 and 110; hope you can help me with this. – Alan Mar 11 '16 at 18:35