A surprising property of nonlocal operators: the deregularising effect of nonlocal elements in convolution differential equations

Abstract

We consider nonlocal differential equations with convolution coefficients of the form−M(a*|u|q)(1)μ(t)u″(t)=λft,u(t), t∈(0,1), $$-M\left(\left(a {\ast} \vert u{\vert }^{q}\right)\left(1\right)\mu \left(t\right)\right){u}^{{\prime\prime}}\left(t\right)=\lambda f\left(t,u\left(t\right)\right)\text{,\,}t\in \left(0,1\right),$$ where q > 0, subject to given boundary data. The function μ∈C[0,1] $\mu \in \mathcal{C}\left(\left[0,1\right]\right)$ modulates the strength of the nonlocal element. We demonstrate that the nonlocality has a strong deregularising effect in the specific sense that nonexistence theorems for this problem are directly affected by the magnitude of the function μ. A specific example illustrates the application of the nonexistence results presented herein.

Keywords

• nonlocal differential equation
• nonexistence
• finite convolution
• primary: 33b15
• 34b10
• 34b18
• 42a85
• 44a35
• secondary: 26a33
• 47h30