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Recently I've conisdered a functional derivative estimate on the first spatial derivative of bounded classical solutions $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ to $$ u_t - u_{xx} - f(u) = 0 \ \ \ (x, …

I am not convinced the argument works under the current specifications you have given (it can, but you need to make them). You claim that the last two inequalities lead to the desired result (I assume …

Smoothness on $G$ may not be enough to ensure that a maximum principle can be obtained. If $G$ or $G_x$ blows up as $|x|\to\infty$ then I anticipate a problem. I advise looking at Protter and Weinberg …

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}\ …

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}\ …

I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial …

If you wish to consider continuous dependence for your problem, you must further specify the type of solutions you are interested in (i.e. solutions that satisfy a particular bound), for otherwise non …