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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 Lr 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 Lp 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 Lr 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 Lp 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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