Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on $\mathbb{R}\times (0,T]$) to a Cauchy problem for a reaction-diffusion equation $$ u_t - u_{xx} = f(u) \ \ \ \text{on}\ \ \ \mathbb{R}\times (0,T]$$ $$ u(x,0) = u_0(x)\ \ \ \forall x\in\mathbb{R}.$$ Here the initial data $u_0:\mathbb{R}\to\mathbb{R}$ is bounded and continuous and the reaction function $f:\mathbb{R}\to\mathbb{R}$ is bounded and continuous, but not locally Holder continuous of any degree. Are there any results guaranteeing the (local) existence of solutions to this problem? Alternatively, are there any results available to suggest that there should not be a local existence result for this problem?


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