Evolution equations with nonlocal initial conditions and superlinear growth. (English) Zbl 1485.35273
Summary: We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case, we are able to show the existence of at least one periodic solution on the half line.
MSC:
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 35K58 | Semilinear parabolic equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 47H06 | Nonlinear accretive operators, dissipative operators, etc. |
| 47H10 | Fixed-point theorems |
Keywords:
approximation solvability method; superlinear nonlinearity; Nemytskii operator; nonlocal conditionsReferences:
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