Global Well-Posedness in Variable Exponent Besov-Morrey Spaces for the Fractional Porous Medium Equation

Abstract

This paper is concerned with the study of the fractional porous medium equation in variable exponent Besov-Morrey spaces. In the case 1 < β <= 2, by using the Chemin mono-norm method, we prove global well-posedness for small initial data in the critical variable exponent Besov-Morrey spaces N_{r(·),q(·),h}^{2-2m-β+d/q(·)}(R^d)  with  (1-\varepsilon)/2 <m<1+d/(2q(·)),  0<\varepsilon< β-1,  1 ≤ r(·) ≤ q(·) < ∞  and  1 ≤ h ≤ ∞. In the limit case β=1,  we establish the global well-posedness for small initial data in  N_{r(·),q(·),1}^{1-2m+d/q(·)}(R^d)  with  1/2 <m<1+d/(2q(·))  and   1 ≤ r(·) ≤ q(·) < ∞.