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

We consider nonlocal differential equations with convolution coefficients of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mo>−</m:mo> <m:mi>M</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mspace width="0.17em"/> <m:mo>*</m:mo> <m:mspace width="0.17em"/> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>μ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>″</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mi>f</m:mi> <m:mfenced close=")" open="("> <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>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mtext>, </m:mtext> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>0,1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:math> -M((a u q)(1)μ (t))u^(t)=λ f(t,u(t)),\,t∈ (0,1), where q > 0, subject to given boundary data. The function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>μ</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">C</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mn>0,1</m:mn> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:mfenced> </m:math> μ ∈ C([0,1]) 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.