• Previous Article
    Random attractors for stochastic parabolic equations with additive noise in weighted spaces
  • CPAA Home
  • This Issue
  • Next Article
    Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction
May 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 PratoS. 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 NeervenM. 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 NeervenM. C. Veraar and L. Weis, Stochastic maximal $ L_p$-regularity, Ann. Probab., 40 (2012), 788-812.

[29]

J. M. A. M. van NeervenM. 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 PratoS. 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 NeervenM. 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 NeervenM. C. Veraar and L. Weis, Stochastic maximal $ L_p$-regularity, Ann. Probab., 40 (2012), 788-812.

[29]

J. M. A. M. van NeervenM. 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.

[1]

Tôn Việt Tạ. Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4507-4542. doi: 10.3934/dcds.2017193

[2]

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130

[3]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124

[4]

Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137

[5]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

[6]

Genghong Lin, Zhan Zhou. Homoclinic solutions of discrete $ \phi $-Laplacian equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1723-1747. doi: 10.3934/cpaa.2018082

[7]

Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109

[8]

Chengxiang Wang, Li Zeng, Wei Yu, Liwei Xu. Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction. Inverse Problems & Imaging, 2018, 12 (3) : 545-572. doi: 10.3934/ipi.2018024

[9]

Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058

[10]

Renato Huzak. Cyclicity of degenerate graphic $DF_{2a}$ of Dumortier-Roussarie-Rousseau program. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1305-1316. doi: 10.3934/cpaa.2018063

[11]

Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1441-1460. doi: 10.3934/dcds.2018059

[12]

Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068

[13]

Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2018087

[14]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[15]

Wenqiang Zhao. Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2018065

[16]

Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $ S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007

[17]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129

[18]

Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062

[19]

James Tanis. Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 989-1006. doi: 10.3934/dcds.2018042

[20]

Valeria Banica, Luis Vega. Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1317-1329. doi: 10.3934/cpaa.2018064

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (106)
  • HTML views (316)
  • Cited by (0)

Other articles
by authors

[Back to Top]