Article Contents
Article Contents

# Existence results for linear evolution equations of parabolic type

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.

Mathematics Subject Classification: Primary: 47D06, 60H15; Secondary: 35R60.

 Citation:

•  J. M. Ball , Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977) , 370-373. Z. Brzeźniak  and  E. Hausenblas , Maximal regularity for stochastic convolutions driven by Levy processes, Probab. Theory Related Fields, 145 (2009) , 615-637. G. Da Prato  and  P. Grisvard , Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal., 58 (1984) , 107-124. G. Da Prato , S. Kwapien  and  J. Zabczyk , Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastic, 23 (1987) , 1-23. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second edition, Cambridge University Press, Cambridge, 2014. 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. A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel-Dekker, 1999. M. Hairer, An introduction to stochastic PDEs, arXiv e-prints (2009), arXiv: 0907.4178. 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. 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. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. R. Mikulevicius , On the Cauchy problem for parabolic SPDEs in Hölder classes, Ann. Probab., 28 (2000) , 74-103. C. Mueller  and  D. Nualart , Regularity of the density for the stochastic heat equation, Electron. J. Probab., 13 (2008) , 2248-2258. E. Pardoux  and  T. Zhang , Absolute continuity of the law of the solution of a parabolic SPDE, J. Functional Anal., 13 (2008) , 2248-2258. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. B. L. Rozovskii, Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Kluwer Academic Publishers Group, Dordrecht, 1990. E. Sinestrari , On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985) , 16-66. T. Shiga , Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Can. J. Math., 46 (1994) , 415-437. H. Tanabe , Remarks on the equations of evolution in a Banach space, Osaka J. Math., 12 (1960) , 145-166. H. Tanabe , Note on singular pertubation for abstract differential equations, Osaka J. Math., 1 (1964) , 239-252. H. Tanabe, Equation of Evolution, Iwanami (in Japanese), 1975. English translation, Pitman, 1979. H. Tanabe, Functional Analytical Methods for Partial Differential Equations, Marcel-Dekker, 1997. T. V. Tạ , Regularity of solutions of abstract linear evolution equations, Lith. Math. J., 56 (2016) , 268-290. 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. T. V. Tạ, Stochastic parabolic evolution equations in M-type 2 Banach spaces, arXiv e-prints, (2015), arXiv: 1508.07340. 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). 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. J. M. A. M. van Neerven , M. C. Veraar  and  L. Weis , Stochastic maximal $L_p$-regularity, Ann. Probab., 40 (2012) , 788-812. J. M. A. M. van Neerven , M. C. Veraar  and  L. Weis , Maximal $γ$-regularity, J. Evol. Equ., 15 (2015) , 361-402. M. C. Veraar , Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ., 10 (2010) , 85-127. 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. A. Yagi , Fractional powers of operators and evolution equations of parabolic type, Proc. Japan Acad. Ser. A Math. Sci., 64 (1988) , 227-230. A. Yagi , Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkcial. Ekvac., 32 (1989) , 107-124. A. Yagi , Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Ⅱ, Funkcial. Ekvac., 33 (1990) , 139-150. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010.