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

Abadias, Luciano; Alvarez, Edgardo; Lizama, Carlos

J. Integral Equations Appl. 30, No. 2, 219-256 (2018).

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 \(\theta\).

Reviewer: Gustaf Gripenberg (Helsingfors)

MSC:

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

Keywords:

Volterra integral equations; local and global; extension and blow up; mild solutions; \(\epsilon\)-regularity

References:

[1]	L. Abadias, C. Lizama and P.J. Miana, Sharp extensions and algebraic properties for solutions families of vector-valued differential equations, Banach J. Math. Anal. 10 (2016), 169–208. · Zbl 1380.47034
[2]	J.M. Arrieta and A.N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc. 352 (1999), 285–310. · Zbl 0940.35119
[3]	E. Bajlekova, The abstract Cauchy problem for the fractional evolution equation, Fract. Calc. Appl. Anal. 1 (1998), 255–270.
[4]	H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996), 277–304. · Zbl 0868.35058
[5]	T. Burton, Volterra Integral and differential equations, Elsevier, Amsterdam, 2005. · Zbl 1075.45001
[6]	Ph. Clément and G. Da Prato, Existence and regularity results for an integral equation with infinite delay in a Banach space, Int. Eq. Oper. Th. 11 (1988), 480–500. · Zbl 0668.45010
[7]	Ph. Clément and E. Mitidieri, Qualitative properties of solutions of Volterra equations in Banach spaces, Israel J. Math. 64 (1988), 1–24. · Zbl 0675.45016
[8]	B. De Andrade, A.N. Carvalho, P.M. Carvalho-Neto and P. Marín-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topol. Meth. Nonlin. Anal. 45 (2015), 439–467. · Zbl 1368.34018
[9]	K.J. Engel and R. Nagel, One-Parameter semigroups for linear evolution equations, Grad. Texts Math. 194, Springer, New York, 2000. · Zbl 0952.47036
[10]	A. Friedman, On integral equations of Volterra type, J. Anal. Math. 11 (1963), 381–413. · Zbl 0134.31502
[11]	G. Gripenberg, On Volterra equations of the first kind, Int. Eq. Oper. Th. 3 (1980), 473–488. · Zbl 0445.45001
[12]	G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations, in, Encyclopedia of mathematics and its applications34, Cambridge University Press, Cambridge, 1990. · Zbl 0695.45002
[13]	B.H. Guswanto and T. Suzuki, Existence and uniqueness of mild solutions for fractional semilinear differential equations, Electr. J. Diff. Eq. 168 (2015), 1–16. · Zbl 1329.34008
[14]	M. Haase, The functional calculus for sectorial operators, in Operator theory: Advances and applications169, Birkhauser, Basel, 2006. · Zbl 1101.47010
[15]	A. Jawahdou, Mild solutions of functional semilinear evolution Volterra integrodifferential equations on an unbounded interval, Nonlin. Anal. 74 (2011), 7325–7332. · Zbl 1235.34203
[16]	A.A.Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006. · Zbl 1092.45003
[17]	J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in \(\mathbb{R}^d\), Math. Ann. 366 (2016), 941–979. · Zbl 1354.35178
[18]	K. Li and J. Peng, Fractional resolvents and fractional evolution equations, Appl. Math. Lett. 25 (2012), 808–812. · Zbl 1253.34016
[19]	C. Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl. 243 (2000), 278–292. · Zbl 0952.45005
[20]	C. Lizama and G. N’Guérékata, Mild solutions for abstract fractional differential equations, Appl. Anal. 92 (2013), 1731–1754. · Zbl 1276.34004
[21]	Z.D. Mei, J.G. Peng and Y. Zhang, A characteristic of fractional resolvents, Fract. Calc. Appl. Anal. 16 (2013), 777–790. · Zbl 1314.34022
[22]	Z. Ouyang, Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comp. Math. Appl. 61 (2011), 860–870. · Zbl 1217.35206
[23]	J. Prüss, Evolutionary integral equations and applications, Birkhauser/Springer, Basel, 2012.
[24]	C. Roberts, Characterizing the blow-up solutions for nonlinear Volterra integral equations, in Proceedings of the Second World Congress of Nonlinear Analysts, Part 2, Nonlin. Anal. 30 (1997), 923–933. · Zbl 0891.45003
[25]	F.B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979), 277–296. · Zbl 0419.47031
[26]	——–, Local existence and nonexistence for semilinear parabolic equations in \(L_p\), Indiana Univ. Math. J. 29 (1980), 79–102. · Zbl 0443.35034