Let $$u_t = au_{xx}+ cu_x, a>0$$ With $$u(x,0)=f(x), u(0,t)=u(\pi,t)=0$$
Find a bound for $c$ so that the solution doesn't grow for each $f(x)$.
My attempt (all integrals are from $0$ to $\pi$):
$$uu_t = auu_{xx}+ cuu_x$$
$$0.5||u||^2_t = \int uu_t dx= a\int uu_{xx}dx+ c\int uu_x dx= auu_x|_0^\pi - a\int (u_x)^2dx + \frac{c}{2} \int(u^2)_xdx $$
$$=0 - a\int (u_x)^2dx + \frac{c}{2} u^2|_0^\pi = - a\int (u_x)^2dx + \frac{c}{2}(u^2(\pi ,t) - u^2(0,t))=-a \int (u_x)^2dx<0$$
Does that mean that the solution dos not grow for every $c$? It seems that I'm missing something.