Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions. (English) Zbl 1391.34101
Summary: This paper is devoted to study the existence and regularity of mild solutions in some interpolation spaces for a class of functional partial differential equations with nonlocal initial conditions. The linear part is assumed to be a sectorial operator in Banach space \(X\). The fractional power theory and \(\alpha\)-norm are used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. Moreover, we present an example to illustrate the application of main results.
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34K30 | Functional-differential equations in abstract spaces |
| 35D35 | Strong solutions to PDEs |
Keywords:
functional partial differential equation; nonlocal condition; strong solution; analytic compact semigroup; fractional power operator; \(\alpha\)-normReferences:
| [1] | Deng K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 1993, 179, 630-637. · Zbl 0798.35076 |
| [2] | Byszewski L., Existence and uniqueness of a classical solution to a functional differential abstract nonlocal Cauchy problem, J. Math. Appl. Stoch. Anal., 1999, 12, 91-97. · Zbl 0934.34067 |
| [3] | Boucherif A., Semilinear evolution inclutions with nonlocal conditions, Appl. Math. Letters, 2009, 22, 1145-1149. · Zbl 1173.34338 |
| [4] | Byszewski L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Appl. Anal., 1991, 12, 494-505. · Zbl 0748.34040 |
| [5] | Ezzinbi K., Fu X., Hilal K., Existence and regularity in the α-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 2007, 67, 1613-1622. · Zbl 1119.35105 |
| [6] | Xiao T.J., Liang J., Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear Anal., 2005, 63, 225-232. · Zbl 1159.35383 |
| [7] | Fu X., Ezzinbi K., Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal., 2003, 54, 215-227. · Zbl 1034.34096 |
| [8] | Bahuguna D., Existence, uniqueness and regularity of solutions to semilinear nonlocal functional differential problems, Nonlinear Anal., 2004, 57, 1021-1028. · Zbl 1065.34078 |
| [9] | Bahuguna D., Agarwal S., Approximations of solutions to neutral didderential equations with nonlocal history conditions, J. Math. Anal. Appl., 2006, 317, 583-602. · Zbl 1101.34063 |
| [10] | Chang J.C., Liu H., Existence of solutions in some interpolation spaces for a class of semilinear evolutions with nonlocal initial conditions, J. Funct. Spaces Appl., 2013, Art. ID 451252, 11 pp. · Zbl 1446.35064 |
| [11] | Liang J., Van Casteren J., Xiao T.J., Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear Anal., 2002, 50, 173-189. · Zbl 1009.34052 |
| [12] | Liu H., Chang J.C., Existence for a class of partial differential equations with nonlocal conditions, Nonlinear Anal., 2009, 70, 3076-3083. · Zbl 1170.34346 |
| [13] | Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Berlin: Springer-Verlag, 1983. · Zbl 0516.47023 |
| [14] | Travis C.C., Webb G.F., Existence, stability, and compactness in the α-norm for partial functional differential equations, Trans. Amer. Math. Soc., 1978, 240, 129-143. · Zbl 0414.34080 |
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