Conclusion
The results presented in § 1 and § 2 can serve as a basis for further study of equations like (1.1), (2.1)–(2.3). For example, using the obtained estimates for a solution u(x,t) together with the well-known estimates for solutions to the Cauchy problem or the maximum principle for parabolic equations [6, 7], we can easily obtain estimates for the derivativesu t (x, t),u tt (x, t), etc., as well as estimates for the derivatives with respect to the space variables.
Concluding the article, we note that, in our opinion, together with the questions of existence and nonexistence of smooth solutions it is worthwhile to study some questions that concern qualitative properties of solutions to the considered equations, for example the questions of localization of solutions and some other questions.
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Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 100–1005, September–October, 1994.
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Kozhanov, A.I. Parabolic equations with nonlocal nonlinear source. Sib Math J 35, 945–956 (1994). https://doi.org/10.1007/BF02104572
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DOI: https://doi.org/10.1007/BF02104572