I am relatively familiar with the heat equation and using Schauder estimates when studying the regularity of the solution, i.e. for some given $g$ in some function space finding an $f$ which solves $$(\partial_{xx}-\partial_{t})f=g$$ $$f(0)=f_0.$$ This equation can be solved by convolving with the fundamental solution which is a gaussian kernel, i.e. one can use semi-group theory.
I was wandering how things change if the minus in the equation is replaced by a plus, i.e. if we study $$(\partial_{xx}+\partial_{t})f=g$$ $$f(0)=f_0.$$ Can we still derive Schauder-like estimates? Or is the equation fundamentally different?
