Regularity properties of mild solutions for a class of Volterra equations with critical nonlinearities. (English) Zbl 1404.45001

The authors study the nonlinear Volterra equation \[ u(t)= \int_0^t a(t-s)[Au(s)+ f(s,u(s))]\, ds + u_0, \quad t\geq 0, \] in a Banach space \(X\) where \(A\) is a densely defined sectorial linear operator that generates an integral resolvent \(S(t)\) with respect to the kernel \(a\in L^1_{\text{loc}}(\mathbb R_+)\) for which there is a function \(b\in L^1_{\text{loc}}(\mathbb R_+)\) such that \(\int_0^t a(t-s)b(s)\, ds =1\) when \(t>0\). The function \(f\) is assumed to be locally Lipschitz continuous with respect to the second variable, uniformly with respect to the first one. The authors prove the existence of a unique local mild solution, i.e., a function \(u\) satisfying \(u(t)= R(t)u_0 + \int_0^tS(t-s)f(s,u(s))\, ds\) (where \(R(t)x= \int_0^t b(t-s)S(s)x\, ds\)) and then show that this local solution can be extended to a maximal one that blows up unless it is a global one. For the case where \(a(t)= t^{\alpha-1}e^{-\delta t}/\Gamma(\alpha)\) and \(f\) satisfies certain additional assumptions, the authors establish some regularity results for the solution, e.g., that \(u(t)\in D((-A)^{1+\theta})\) for certain positive values of \(\theta\).


45D05 Volterra integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
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