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Sharp pointwise estimates for the gradients of solutions to linear parabolic second-order equation in the layer. (English) Zbl 1484.35094

Summary: We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second-order equations with real constant coefficients in the layer \(\mathbb{R}^{n+1}_T = \mathbb{R}^n \times (0,T)\), where \(n \geq 1\) and \(T < \infty\). The homogeneous equation is considered with initial data in \(L^p(\mathbb{R}^n)\), \(1 \leq p \leq \infty\). For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to \(f \in L^p(\mathbb{R}^{n+1}_T) \cap C^{\alpha}(\overline{\mathbb{R}^{n+1}_T})\), \(p>n + 2\) and \(\alpha \in (0,1)\). Explicit formulas for the sharp coefficients in pointwise estimates for the length of the gradient to solutions to these problems are obtained.

MSC:

35B45 A priori estimates in context of PDEs
35K15 Initial value problems for second-order parabolic equations
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References:

[1] Polyanin, AD; Zaitsev, VF., Handbook of nonlinear partial differential equations. (2004), Chapman and Hall/CRC Press · Zbl 1053.35001
[2] Tikhonov, AN; Samarskii, AA., Equations of mathematical physics. (1990), Dover Publications
[3] Kresin, G.; Maz’ya, V., Maximum principles and sharp constants for solutions of elliptic and parabolic systems, 183 (2012), Providence, Rhode Island: Amer. Math. Soc., Providence, Rhode Island · Zbl 1255.35001
[4] Kresin, G.; Maz’ya, V., Sharp estimates for the gradient of solutions to the heat equation, Algebra i Analiz, 31, 3, 136-153 (2019) · Zbl 1446.35047
[5] Kresin, G.; Maz’ya, V., Optimal estimates for derivatives of solutions to Laplace, LamĂ© and Stokes equations, J. Math. Sci., New York, 196, 3, 300-321 (2014) · Zbl 1302.31008 · doi:10.1007/s10958-014-1660-2
[6] Kresin, G.; Maz’ya, V., Sharp real-part theorems. A unified approach, 1903 (2007), Berlin: Springer, Berlin · Zbl 1117.30001
[7] Lancaster, P., Theory of matrices (1969), New York: Academic Press, New York · Zbl 0186.05301
[8] Hunter, JK., Notes on partial differential equations (2014), Davis: University of California, Davis
[9] Oleinik, OA., Lectures on partial differential equations (2005), BINOM: Moscow, BINOM
[10] Shubin, MA., Lectures on equations of mathematical physics (2003), MCNMO: Moscow, MCNMO
[11] Vladimirov, VS., Equations of mathematical physics (1984), Moscow: Mir Publ., Moscow
[12] Friedman, A., Partial differential equations of parabolic type (1983), Malabar, Florida: R.E. Krieger Publ. Comp., Malabar, Florida
[13] Vulich, BZ., A short course in the theory of functions of the real variable (1973), Moscow: Nauka, Moscow
[14] Gradshtein, IS, Ryzhik, IM, Jeffrey, A, (editor). Table of integrals, series and products. 5th ed. New York: Academic Press; 1994. · Zbl 0918.65002
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