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Variable exponent spaces have found interesting applications in real worldproblems. Recently, there have been considerable interest in utilizingvariational and evolution problems based on variable exponents for imagingapplications. The main classes of partial differential equations (PDEs) relatedto the variable exponents involve the $p(\cdot)$-Laplacian. In imagingapplications, the variable exponent can approach the critical value $1$, andthis poses unique challenges in proving existence of solutions, which have notbeen mastered earlier. In this work, we develop some additional functionalframework to study the time-dependent parabolic flows with critical variableexponents. Specifically, we consider bounded vectorial partial variation (BVPV)space and its variable exponent counterpart. We prove the existence of weaksolutions of critical vectorial $p(t,x)$-Laplacian flow in our variableexponent space. For non-time-dependent variable exponent based criticalvectorial $p(x)$-Laplacian flow we obtain a semigroup solution. The results arenew even in the scalar case. This is a theory-oriented draft, and the fullpaper will provide detailed experimental results on color image restorationusing various example for the variable exponents and compare them traditionalPDE based image processing procedures. Our results will indicate theapplicability of variable exponent Laplacian flows in image processing ingeneral and image restoration in particular.

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