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I was wondering if anybody knows (and can give me a reference, please) if the PDE below has a unique weak solution. I can only find the result if we consider homogeneous Dirichlet boundary conditions, but not for non-homogeneous as I need.

Many thanks!

$$ \partial_t \rho(t,u) = \partial_u^2\rho(t,u)-E\partial_u f(\rho(t,u))\\ \rho(t,0) = \alpha\; \rho(t,1) = \beta, \quad 0\leq t\leq T,\\ \rho(0,u) = \rho_0(u)\; \quad 0\leq u \leq 1 $$ with $f(x)=x(1-x)$ and $\alpha$, $\beta$ and $E$ constants;

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Try to substract $g(x)=\alpha-x(\alpha-\beta)$. Now the dirichlet conditions of the new unknown $h=\rho-g$ are homogeneous. The new problem is $$ \partial_t h=\partial_u^2h-E\partial_u f(h)+F(g). $$ with initial data $h_0=\rho_0-g$. Now I think that the usual theory applies straightforwardly.

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