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

Existence results for linear evolution equations of parabolic type

Tôn Việt Tạ

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.

Keywords: Parabolic evolution equations, maximal regularity, cylindrical Wiener process, $ \gamma$-radonifying operators, UMD Banach spaces of type 2, strict solutions.
Mathematics Subject Classification: Primary: 47D06, 60H15; Secondary: 35R60.
Citation: Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure and 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.

show all references

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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