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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?

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    $\begingroup$ you use the method of eigenfunction expansion twice I believe $\endgroup$
    – user3417
    Oct 17, 2018 at 22:07
  • $\begingroup$ @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. $\endgroup$
    – Oleg
    Oct 18, 2018 at 1:36
  • $\begingroup$ it's apparently called gronwells inequality $\endgroup$
    – user3417
    Oct 18, 2018 at 1:36
  • $\begingroup$ math.ucdavis.edu/~hunter/pdes/ch6.pdf it's on page 14 $\endgroup$
    – user3417
    Oct 18, 2018 at 1:37
  • $\begingroup$ I think the other theorem is picard lindelof en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem $\endgroup$
    – user3417
    Oct 18, 2018 at 3:52

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