Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture

In this paper, we consider the general dual fractional parabolic problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:msubsup><m:mrow><m:mi>∂</m:mi></m:mrow><m:mrow><m:mi>t</m:mi></m:mrow><m:mrow><m:mi>α</m:mi></m:mrow></m:msubsup><m:mi>u</m:mi><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>t</m:mi></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mo>+</m:mo><m:mi mathvariant="script">L</m:mi><m:mi>u</m:mi><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>t</m:mi></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>f</m:mi><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mi>t</m:mi><m:mo>,</m:mo><m:mi>u</m:mi><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>t</m:mi></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mspace width="0.3333em"/><m:mspace width="0.3333em"/><m:mtext>in</m:mtext><m:mspace width="0.3333em"/><m:mspace width="0.3333em"/><m:msup><m:mrow><m:mi mathvariant="double-struck">R</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:msup><m:mo>×</m:mo><m:mi mathvariant="double-struck">R</m:mi><m:mo>.</m:mo></m:math> ∂ ₜαu(x,t)+Lu(x,t)=f(t,u(x,t)) in Rⁿ×R. We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f . Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:mi mathvariant="script">L</m:mi></m:math> L and the Marchaud time derivative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:msubsup><m:mrow><m:mi>∂</m:mi></m:mrow><m:mrow><m:mi>t</m:mi></m:mrow><m:mrow><m:mi>α</m:mi></m:mrow></m:msubsup></m:math> ∂ ₜα , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:mi mathvariant="script">L</m:mi></m:math> L , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"><m:mi mathvariant="script">L</m:mi></m:math> L , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.