On well-posedness for parabolic Cauchy problems of Lions type with rough initial data

Abstract We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of p for which tempered distributions in homogeneous Hardy–Sobolev spaces $$\dot{H}^{s,p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:math> with regularity index $$s \in (-1,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are initial data. Source terms of Lions’ type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $$\dot{B}^{s}_{p,p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:math> .