Consider a semilinear heat equation on $[0,1]$ $$ \frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+b(t,x,u(t,x)). $$ We assume here the periodic boundary conditions, that is $u(0,t)=u(1,t)$ and $\frac{\partial u}{\partial x}(0,t)=\frac{\partial u}{\partial x}(1,t)$.
Suppose that the function $b$ is nice i.e that it is smooth in all arguments and some growth condition holds
$$y b(t,x,y)\le C_1+C_2y^2.$$
How can we prove that this equation has a unique solution? I couldn't find an appropriate theorem in Evans' book, but maybe I should look somewhere else?