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On Solvability of Parabolic Equations with Essentially Nonlinear Differential-Difference Operators

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Abstract

We consider the first mixed boundary value problem for a nonlinear differential-difference parabolic equation. We give some sufficient conditions for the nonlinear differential-difference operator to be radially continuous and coercive as well as has the property of (V,W)-semibounded variation (in this case we provide the algebraic condition of strong ellipticity for an essentially nonlinear differential-difference operator). We also justify the existence theorems for a generalized solution.

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Notes

  1. The coercivity property is usually employed to justify the pseudomonotonicity of operators on \( W \) in the case when we use the ellipticity condition instead of the algebraic condition of the strong ellipticity. Note also that to localize a solution to the problem on some set, we often refer to the Acute Angle Lemma. The coercivity property is very convenient but not the only option.

  2. Note that (A3) contains only higher-order derivatives. Hence, an estimate of the type (A2) cannot be obtained on using this estimate.

  3. Cf. the estimate of Lemma 4.2 corresponding to condition (A4) for \( p\in[2,\infty) \).

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Funding

The authors were supported by the Ministry of Science and Higher Education of the Russian Federation (Megagrant no. 075–15–2022–1115), People’s Friendship University of Russia.

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Authors and Affiliations

  1. Federal Research Center Computer Science and Control of the Russian Academy of Sciences, Moscow, 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 Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 5, pp. 1094–1113. https://doi.org/10.33048/smzh.2023.64.515

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Solonukha, O.V. On Solvability of Parabolic Equations with Essentially Nonlinear Differential-Difference Operators. Sib Math J 64, 1237–1254 (2023). https://doi.org/10.1134/S0037446623050154

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  • DOI: https://doi.org/10.1134/S0037446623050154

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