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Solvability of boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in classes of functions growing at infinity

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Abstract

For divergent elliptic equations with the natural energetic spaceW mp (Ω),m≥1,p>2, we prove that the Dirichlet problem is solvable in a broad class of domains with noncompact boundaries if the growth of the right-hand side of the equation is determined by the corresponding theorem of Phragmén-Lindelöf type. For the corresponding parabolic equation, we prove that the Cauchy problem is solvable for the limiting growth of the initial function % MathType!MTEF!2!1!+-

$$u_0 (x) \in L_{2.loc} (R^n ): \int\limits_{|x|< \tau } {u_0^2 dx \leqslant c\tau ^{n + 2mp/(p - 2)} \forall \tau< \infty } $$

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk

    A. E. Shishkov

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  1. A. E. Shishkov
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 277–289, February, 1995.

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Shishkov, A.E. Solvability of boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in classes of functions growing at infinity. Ukr Math J 47, 326–340 (1995). https://doi.org/10.1007/BF01056723

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  • DOI: https://doi.org/10.1007/BF01056723

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