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2018, 17(3): 751-785. doi: 10.3934/cpaa.2018039

## Existence results for linear evolution equations of parabolic type

 Center for Promotion of International Education and Research, Faculty of Agriculture, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan

Received  April 2017 Revised  August 2017 Published  January 2018

Fund Project: This research was partially supported by Q-PIT, Kyushu University

We study a stochastic parabolic evolution equation of the form $dX+AXdt = F(t)dt+G(t)dW(t)$ in Banach spaces. Existence of mild and strict solutions and their space-time regularity are shown in both the deterministic and stochastic cases. Abstract results are applied to a nonlinear stochastic heat equation.

Citation: Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039
##### References:
 [1] J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373. [2] Z. Brzeźniak and E. Hausenblas, Maximal regularity for stochastic convolutions driven by Levy processes, Probab. Theory Related Fields, 145 (2009), 615-637. [3] G. Da Prato and P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal., 58 (1984), 107-124. [4] G. Da Prato, S. Kwapien and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastic, 23 (1987), 1-23. [5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second edition, Cambridge University Press, Cambridge, 2014. [6] G. Da Prato and A. Lunardi, Maximal regularity for stochastic convolutions in $L_p$ spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (1998), 25-29. [7] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel-Dekker, 1999. [8] M. Hairer, An introduction to stochastic PDEs, arXiv e-prints (2009), arXiv: 0907.4178. [9] N. V. Krylov, An analytic approach to SPDEs, in stochastic partial differential equations: Six perspectives, Math. Surveys Monogr. Amer. Math. Soc., 64 (1999), 185-242. [10] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal., 31 (1999), 19-33. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. [12] R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Hölder classes, Ann. Probab., 28 (2000), 74-103. [13] C. Mueller and D. Nualart, Regularity of the density for the stochastic heat equation, Electron. J. Probab., 13 (2008), 2248-2258. [14] E. Pardoux and T. Zhang, Absolute continuity of the law of the solution of a parabolic SPDE, J. Functional Anal., 13 (2008), 2248-2258. [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. [16] B. L. Rozovskii, Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Kluwer Academic Publishers Group, Dordrecht, 1990. [17] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), 16-66. [18] T. Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Can. J. Math., 46 (1994), 415-437. [19] H. Tanabe, Remarks on the equations of evolution in a Banach space, Osaka J. Math., 12 (1960), 145-166. [20] H. Tanabe, Note on singular pertubation for abstract differential equations, Osaka J. Math., 1 (1964), 239-252. [21] H. Tanabe, Equation of Evolution, Iwanami (in Japanese), 1975. English translation, Pitman, 1979. [22] H. Tanabe, Functional Analytical Methods for Partial Differential Equations, Marcel-Dekker, 1997. [23] T. V. Tạ, Regularity of solutions of abstract linear evolution equations, Lith. Math. J., 56 (2016), 268-290. [24] T. V. Tạ, Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: strict solutions and maximal regularity, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4507-4542. [25] T. V. Tạ, Stochastic parabolic evolution equations in M-type 2 Banach spaces, arXiv e-prints, (2015), arXiv: 1508.07340. [26] T. V. Tạ, Y. Yamamoto and A. Yagi, Strict solutions to stochastic linear evolution equations in M-type 2 Banach spaces, Funkc. Ekvacioj. (in press) (arXiv: 1508.07431). [27] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993. [28] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic maximal $L_p$-regularity, Ann. Probab., 40 (2012), 788-812. [29] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Maximal $γ$-regularity, J. Evol. Equ., 15 (2015), 361-402. [30] M. C. Veraar, Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ., 10 (2010), 85-127. [31] J. B. Walsh, An Introduction to Stochastic Partial Differential Equations, École d'été de probabilités de Saint-Flour, XIV-1984,265-439, Lecture Notes in Mathematics 1180, Springer, Berlin, 1986. [32] A. Yagi, Fractional powers of operators and evolution equations of parabolic type, Proc. Japan Acad. Ser. A Math. Sci., 64 (1988), 227-230. [33] A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkcial. Ekvac., 32 (1989), 107-124. [34] A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Ⅱ, Funkcial. Ekvac., 33 (1990), 139-150. [35] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010.

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##### References:
 [1] J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373. [2] Z. Brzeźniak and E. Hausenblas, Maximal regularity for stochastic convolutions driven by Levy processes, Probab. Theory Related Fields, 145 (2009), 615-637. [3] G. Da Prato and P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal., 58 (1984), 107-124. [4] G. Da Prato, S. Kwapien and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastic, 23 (1987), 1-23. [5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second edition, Cambridge University Press, Cambridge, 2014. [6] G. Da Prato and A. Lunardi, Maximal regularity for stochastic convolutions in $L_p$ spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (1998), 25-29. [7] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel-Dekker, 1999. [8] M. Hairer, An introduction to stochastic PDEs, arXiv e-prints (2009), arXiv: 0907.4178. [9] N. V. Krylov, An analytic approach to SPDEs, in stochastic partial differential equations: Six perspectives, Math. Surveys Monogr. Amer. Math. Soc., 64 (1999), 185-242. [10] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal., 31 (1999), 19-33. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. [12] R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Hölder classes, Ann. Probab., 28 (2000), 74-103. [13] C. Mueller and D. Nualart, Regularity of the density for the stochastic heat equation, Electron. J. Probab., 13 (2008), 2248-2258. [14] E. Pardoux and T. Zhang, Absolute continuity of the law of the solution of a parabolic SPDE, J. Functional Anal., 13 (2008), 2248-2258. [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. [16] B. L. Rozovskii, Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Kluwer Academic Publishers Group, Dordrecht, 1990. [17] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), 16-66. [18] T. Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Can. J. Math., 46 (1994), 415-437. [19] H. Tanabe, Remarks on the equations of evolution in a Banach space, Osaka J. Math., 12 (1960), 145-166. [20] H. Tanabe, Note on singular pertubation for abstract differential equations, Osaka J. Math., 1 (1964), 239-252. [21] H. Tanabe, Equation of Evolution, Iwanami (in Japanese), 1975. English translation, Pitman, 1979. [22] H. Tanabe, Functional Analytical Methods for Partial Differential Equations, Marcel-Dekker, 1997. [23] T. V. Tạ, Regularity of solutions of abstract linear evolution equations, Lith. Math. J., 56 (2016), 268-290. [24] T. V. Tạ, Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: strict solutions and maximal regularity, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4507-4542. [25] T. V. Tạ, Stochastic parabolic evolution equations in M-type 2 Banach spaces, arXiv e-prints, (2015), arXiv: 1508.07340. [26] T. V. Tạ, Y. Yamamoto and A. Yagi, Strict solutions to stochastic linear evolution equations in M-type 2 Banach spaces, Funkc. Ekvacioj. (in press) (arXiv: 1508.07431). [27] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993. [28] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic maximal $L_p$-regularity, Ann. Probab., 40 (2012), 788-812. [29] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Maximal $γ$-regularity, J. Evol. Equ., 15 (2015), 361-402. [30] M. C. Veraar, Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ., 10 (2010), 85-127. [31] J. B. Walsh, An Introduction to Stochastic Partial Differential Equations, École d'été de probabilités de Saint-Flour, XIV-1984,265-439, Lecture Notes in Mathematics 1180, Springer, Berlin, 1986. [32] A. Yagi, Fractional powers of operators and evolution equations of parabolic type, Proc. Japan Acad. Ser. A Math. Sci., 64 (1988), 227-230. [33] A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkcial. Ekvac., 32 (1989), 107-124. [34] A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Ⅱ, Funkcial. Ekvac., 33 (1990), 139-150. [35] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010.
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