On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing I: Analysis

Variable exponent spaces have found interesting applications in real world problems. In imaging models, the variable exponent can approach the critical value 1 and this poses unique challenges in proving existence of solutions for the corresponding PDEs. In this work, we develop some new functional framework to study time-dependent parabolic variable exponent flows. Specifically, we consider bounded vectorial partial variation (BVPV) space and its variable exponent counterpart. We then prove the existence of weak solutions to the critical vectorial p(t,x)-Laplacian flow in the variable exponent BVPV space. For time-independent critical vectorial p(x)-Laplacian flow we obtain a unique semigroup solution. Our results are in particular valid in the scalar case and solve a long standing open problem.