Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

Md. Rabiul Haque; Takayoshi Ogawa; Ryuichi Sato

doi:10.3934/cpaa.2020031

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February 2020, 19(2): 677-697. doi: 10.3934/cpaa.2020031

Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

Md. Rabiul Haque ^1,, , Takayoshi Ogawa ^2, and Ryuichi Sato ^1,

1.	Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
2.	Mathematical Institute/Research Alliance Center of Mathematical Science, Tohoku University, Sendai 980-8578, Japan

^* Corresponding author

* On leaving from Department of Mathematics, University of Rajshahi-6205, Bangladesh

Received March 2018 Revised May 2019 Published October 2019

We consider the local existence and the uniqueness of a weak solution of the initial boundary value problem to a convection–diffusion equation in a uniformly local function space $ L^r_{{\rm uloc}, \rho}( \Omega) $, where the solution is not decaying at $ |x|\to \infty $. We show that the local existence and the uniqueness of a solution for the initial data in uniformly local $ L^r $ spaces and identify the Fujita-Weissler critical exponent for the local well-posedness found by Escobedo-Zuazua [10] is also valid for the uniformly local function class.

Keywords: Convection-diffusion equation, uniformly local Lebesgue spaces, weak solution.

Mathematics Subject Classification: Primary: 35A01, 35K55, 35D30.

Citation: Md. Rabiul Haque, Takayoshi Ogawa, Ryuichi Sato. Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space. Communications on Pure & Applied Analysis, 2020, 19 (2) : 677-697. doi: 10.3934/cpaa.2020031

References:

[1]	N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225. doi: 10.1016/0022-0396(79)90088-3. Google Scholar
[2]	J. Aguirre, M. Escobedo and E. Zuazua, Self-similar solutions of a convection diffusion equation and related elliptic problems, Comm. Partial Differential Equations, 15 (1990), 139-157. doi: 10.1080/03605309908820681. Google Scholar
[3]	J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Dissipative parabolic equations in locally uniform spaces, Math. Nachr., 280 (2007), 1643-1663. doi: 10.1002/mana.200510569. Google Scholar
[4]	J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234. Google Scholar
[5]	J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal., 56 (2004), 515-554. doi: 10.1016/j.na.2003.09.023. Google Scholar
[6]	H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304. doi: 10.1007/BF02790212. Google Scholar
[7]	J. W. Cholewa and A. Rodríguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 19 (2009), 1995-2037. doi: 10.1142/S0218202509004029. Google Scholar
[8]	G. Duro and E. Zuazua, Large time behavior for convection-diffusion equations in $\mathbb{R}^N$ with periodic coefficients, J. Differential Equations, 167 (2000), 275-315. doi: 10.1006/jdeq.2000.3796. Google Scholar
[9]	G. Duro and E. Zuazua, Large time behavior for convection-diffusion equations in $\mathbb{R}^N$ with asymptotically constant diffusion, Comm. Partial Differential Equations, 24 (1999), 1283-1340. doi: 10.1080/03605309908821466. Google Scholar
[10]	M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $\mathbb{R}^N$, J. Funct. Anal., 100 (1991), 119-161. doi: 10.1016/0022-1236(91)90105-E. Google Scholar
[11]	M. Escobedo and E. Zuazua, Long-time behavior for a convection-diffusion equation in higher dimensions, SIAM J. Math. Anal., 28 (1997), 570-594. doi: 10.1137/S0036141094271120. Google Scholar
[12]	M. Escobedo, J. L. Vázquez and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1993), 43-65. doi: 10.1007/BF00392203. Google Scholar
[13]	M. Escobedo, J. L. Vázquez and E. Zuazua, A diffusion-convection equation in several space dimensions, Indiana Univ. Math. J., 42 (1993), 1413-1440. doi: 10.1512/iumj.1993.42.42065. Google Scholar
[14]	M. Escobedo, J. L. Vázquez and E. Zuazua, Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842. doi: 10.2307/2154744. Google Scholar
[15]	A. Friedman, Partial Differential Equations, Dover, Mineola, 1997. Google Scholar
[16]	R. E. Grundy, Asymptotic solutions of a model nonlinear convective diffusion equation, IMA J. Appl. Math., 31 (1983), 121-137. doi: 10.1093/imamat/31.2.121. Google Scholar
[17]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, Springer, 1983. doi: 10.1007/978-3-642-61798-0. Google Scholar
[18]	M. -H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations, Asymptotic Behavior of Solutions and Self-Similar Solutions, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston Inc., Boston, MA, 2010., doi: 10.1007/978-0-8176-4651-6. Google Scholar
[19]	K. Ishige and R. Sato, Heat equation with a nonlinear boundary condition and uniformly local L^r spaces, Discrete Contin. Dyn. Syst., 36 (2016), 2627-2652. doi: 10.3934/dcds.2016.36.2627. Google Scholar
[20]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. Google Scholar
[21]	G. F. Lu and H. M. Yin, Source-type solutions of heat euations with convection in several variables space, Sci China Math, 56 (2011), 1145-1173. doi: 10.1007/s11425-011-4219-4. Google Scholar
[22]	Y. Maekawa and Y. Terasawa, The Navier-Stokes equations with initial data in uniformly local L^p spaces, Differential Integral Equations, 19 (2006), 369-400. Google Scholar
[23]	J. Matos and P. Souplet, Instantaneous smoothing estimates for the Hermite semigroup in uniformly local spaces and related nonlinear equations, Houston J. Math., 3 (2004), 879-890. Google Scholar
[24]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear Anal., 10 (1986), 299-314. doi: 10.1016/0362-546X(86)90005-2. Google Scholar
[25]	T. Ogawa, Nonlinear Evolutionary Partial Differential Equations, -Method of Real and Harmonic Analysis, Springer-Verlag, to appear. Google Scholar
[26]	F. B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. Google Scholar
[27]	E. Zuazua, Weakly nonlinear large time behavior for scalar convection-diffusion equations, Differential Integral Equations, 6 (1993), 1481-1492. Google Scholar
[28]	E. Zuazua, A dynamical system approach to the self similar large time behavior in scalar convection-diffusion equations, J. Differential Equations, 108 (1994), 1-35. doi: 10.1006/jdeq.1994.1023. Google Scholar

show all references

References:

[1]	N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225. doi: 10.1016/0022-0396(79)90088-3. Google Scholar
[2]	J. Aguirre, M. Escobedo and E. Zuazua, Self-similar solutions of a convection diffusion equation and related elliptic problems, Comm. Partial Differential Equations, 15 (1990), 139-157. doi: 10.1080/03605309908820681. Google Scholar
[3]	J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Dissipative parabolic equations in locally uniform spaces, Math. Nachr., 280 (2007), 1643-1663. doi: 10.1002/mana.200510569. Google Scholar
[4]	J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234. Google Scholar
[5]	J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal., 56 (2004), 515-554. doi: 10.1016/j.na.2003.09.023. Google Scholar
[6]	H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304. doi: 10.1007/BF02790212. Google Scholar
[7]	J. W. Cholewa and A. Rodríguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 19 (2009), 1995-2037. doi: 10.1142/S0218202509004029. Google Scholar
[8]	G. Duro and E. Zuazua, Large time behavior for convection-diffusion equations in $\mathbb{R}^N$ with periodic coefficients, J. Differential Equations, 167 (2000), 275-315. doi: 10.1006/jdeq.2000.3796. Google Scholar
[9]	G. Duro and E. Zuazua, Large time behavior for convection-diffusion equations in $\mathbb{R}^N$ with asymptotically constant diffusion, Comm. Partial Differential Equations, 24 (1999), 1283-1340. doi: 10.1080/03605309908821466. Google Scholar
[10]	M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $\mathbb{R}^N$, J. Funct. Anal., 100 (1991), 119-161. doi: 10.1016/0022-1236(91)90105-E. Google Scholar
[11]	M. Escobedo and E. Zuazua, Long-time behavior for a convection-diffusion equation in higher dimensions, SIAM J. Math. Anal., 28 (1997), 570-594. doi: 10.1137/S0036141094271120. Google Scholar
[12]	M. Escobedo, J. L. Vázquez and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1993), 43-65. doi: 10.1007/BF00392203. Google Scholar
[13]	M. Escobedo, J. L. Vázquez and E. Zuazua, A diffusion-convection equation in several space dimensions, Indiana Univ. Math. J., 42 (1993), 1413-1440. doi: 10.1512/iumj.1993.42.42065. Google Scholar
[14]	M. Escobedo, J. L. Vázquez and E. Zuazua, Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842. doi: 10.2307/2154744. Google Scholar
[15]	A. Friedman, Partial Differential Equations, Dover, Mineola, 1997. Google Scholar
[16]	R. E. Grundy, Asymptotic solutions of a model nonlinear convective diffusion equation, IMA J. Appl. Math., 31 (1983), 121-137. doi: 10.1093/imamat/31.2.121. Google Scholar
[17]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, Springer, 1983. doi: 10.1007/978-3-642-61798-0. Google Scholar
[18]	M. -H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations, Asymptotic Behavior of Solutions and Self-Similar Solutions, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston Inc., Boston, MA, 2010., doi: 10.1007/978-0-8176-4651-6. Google Scholar
[19]	K. Ishige and R. Sato, Heat equation with a nonlinear boundary condition and uniformly local L^r spaces, Discrete Contin. Dyn. Syst., 36 (2016), 2627-2652. doi: 10.3934/dcds.2016.36.2627. Google Scholar
[20]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. Google Scholar
[21]	G. F. Lu and H. M. Yin, Source-type solutions of heat euations with convection in several variables space, Sci China Math, 56 (2011), 1145-1173. doi: 10.1007/s11425-011-4219-4. Google Scholar
[22]	Y. Maekawa and Y. Terasawa, The Navier-Stokes equations with initial data in uniformly local L^p spaces, Differential Integral Equations, 19 (2006), 369-400. Google Scholar
[23]	J. Matos and P. Souplet, Instantaneous smoothing estimates for the Hermite semigroup in uniformly local spaces and related nonlinear equations, Houston J. Math., 3 (2004), 879-890. Google Scholar
[24]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear Anal., 10 (1986), 299-314. doi: 10.1016/0362-546X(86)90005-2. Google Scholar
[25]	T. Ogawa, Nonlinear Evolutionary Partial Differential Equations, -Method of Real and Harmonic Analysis, Springer-Verlag, to appear. Google Scholar
[26]	F. B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. Google Scholar
[27]	E. Zuazua, Weakly nonlinear large time behavior for scalar convection-diffusion equations, Differential Integral Equations, 6 (1993), 1481-1492. Google Scholar
[28]	E. Zuazua, A dynamical system approach to the self similar large time behavior in scalar convection-diffusion equations, J. Differential Equations, 108 (1994), 1-35. doi: 10.1006/jdeq.1994.1023. Google Scholar

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American Institute of Mathematical Sciences

Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

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Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

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