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Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates
Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space
1. | Mathematical Institute, Tohoku University, Sendai 980-8578, Japan |
2. | Mathematical Institute/Research Alliance Center of Mathematical Science, Tohoku University, Sendai 980-8578, Japan |
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 [
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. Friedman, Partial Differential Equations, Dover, Mineola, 1997. |
[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. |
[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. |
[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. |
[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. |
[20] |
T. Kato,
The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal., 58 (1975), 181-205.
doi: 10.1007/BF00280740. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[27] |
E. Zuazua,
Weakly nonlinear large time behavior for scalar convection-diffusion equations, Differential Integral Equations, 6 (1993), 1481-1492.
|
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. Friedman, Partial Differential Equations, Dover, Mineola, 1997. |
[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. |
[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. |
[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. |
[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. |
[20] |
T. Kato,
The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal., 58 (1975), 181-205.
doi: 10.1007/BF00280740. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[27] |
E. Zuazua,
Weakly nonlinear large time behavior for scalar convection-diffusion equations, Differential Integral Equations, 6 (1993), 1481-1492.
|
[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. |
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