Finite and infinite speed of propagation for porous medium equations with fractional pressure

Abstract

We study a porous medium equation with fractional potential pressure: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mo>∂</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mo>⋅</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>m</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>s</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:math> . To be specific, the problem is posed for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>N</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>1</mml:mn> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> . The initial data <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> is assumed to be a bounded function with compact support or fast decay at infinity. We establish the existence of a class of weak solutions for which we determine whether, depending on the parameter m , the property of compact support is conserved in time or not, starting from the result of finite propagation known for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> . We find that when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> the problem has infinite speed of propagation, while for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>∞</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> it has finite speed of propagation. Comparison is made with other nonlinear diffusion models where the results are widely different.