# existence and uniqueness for semilinear heat equation

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?

• you use the method of eigenfunction expansion twice I believe
– user3417
Oct 17, 2018 at 22:07
• @RyanHowe Could you please provide some more details? Or maybe there is a reference, where I can find about it? The point is I do not need to construct the solution, rather just to prove that it exists and unique.
– Oleg
Oct 18, 2018 at 1:36
• it's apparently called gronwells inequality
– user3417
Oct 18, 2018 at 1:36
• math.ucdavis.edu/~hunter/pdes/ch6.pdf it's on page 14
– user3417
Oct 18, 2018 at 1:37
• I think the other theorem is picard lindelof en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
– user3417
Oct 18, 2018 at 3:52