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On the Solvability of Nonlinear Parabolic Functional-Differential Equations with Shifts in the Spatial Variables

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Abstract

The first mixed boundary value problem for a nonlinear functional-differential equation of parabolic type with shifts in the spatial variables is considered. Sufficient conditions are proved under which a nonlinear differential-difference operator is demicontinuous, coercive, and pseudomonotone on the domain of the operator \(\partial_t\). Based on these properties, existence theorems for a generalized solution are proved.

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Notes

  1. Incommensurable shifts were considered only for linear problems. The presence of them can lead to the appearance of a set everywhere dense in the considered domain, which consists of the boundary points. Even in the linear case, the incommensurable shifts seriously complicate the consideration and lead to new effects, for example, the violation of smoothness of the generalized solution on a set everywhere dense in the considered domain, see [10], [18], [19]. In this paper, we consider commensurable shifts only.

  2. In [2], the term pseudomonotone on \(\mathscr D(L)\), where \(L=\partial_t\), is used.

  3. For \(p\in(1,2]\), it suffices to consider \(\widehat p\in(1,2]\). In this case, the coercivity of the problem is ensured by the operator \(\partial_t+A_R\).

  4. In [2], pseudomonotonicity on \(W\) of a differential operator was proved without proving property \((S_+)\) on \(W\) and under a more complicated coercivity condition.

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Acknowledgments

The author thanks the referees for their helpful comments that helped to improve the paper.

Funding

This work was supported by the Ministry for Science and Higher Education of the Russian Federation (megagrant no. 075-15-2022-1115).

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Authors and Affiliations

  1. Federal Research Center “Information and Control”, Moscow, 119333, Russia

    O. V. Solonukha

  2. Рeoples Friendship University of Russia (RUDN University), Moscow, 117198, Russia

    O. V. Solonukha

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  1. O. V. Solonukha
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Correspondence to O. V. Solonukha.

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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 747–763 https://doi.org/10.4213/mzm13781.

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Solonukha, O.V. On the Solvability of Nonlinear Parabolic Functional-Differential Equations with Shifts in the Spatial Variables. Math Notes 113, 708–722 (2023). https://doi.org/10.1134/S0001434623050115

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