Dual fractional parabolic equations with indefinite nonlinearities

In this paper, we consider the following indefinite dual fractional parabolic equation involving the Marchaud fractional time derivative${\partial }_t^{\alpha }u(x,t)+{(-\Delta )}^{s}u(x,t)=a\left(x\right)f\left(u\right(x,t\left)\right)\text{in}{R}^n\times R,$ where $\alpha ,s\in (0,1)$, and the functions a and f are nondecreasing. We prove that there is no positive bounded solutions. To this end, we first show that all positive bounded solutions $u(\cdot ,t)$ must be strictly monotone increasing along the direction determined by $a\left(x\right)$. Then by mollifying the first eigenfunction for fractional Laplacian ${(-\Delta )}^s$ and constructing an appropriate subsolution for the Marchaud fractional operator ${\partial }_t^{\alpha }-1$, we derive a contradiction and thus obtain the non-existence of solutions.

To overcome the challenges caused by the dual non-locality of the operator ${\partial }_t^{\alpha }+{(-\Delta )}^s$, we introduce several new ideas and novel techniques. These novel approaches are not only applicable to the specific problem at hand but can also be extended to address various other fractional problems, be they elliptic or parabolic, including those featuring dual nonlocalities associated with the Marchaud time derivatives.