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November  2013, 33(11&12): 5015-5047. doi: 10.3934/dcds.2013.33.5015

Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions

Alessio Fiscella 1, and  Enzo Vitillaro 2,

1. 

Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy
2. 

Dipartimento di Matematica ed Informatica, Università di Perugia, Via Vanvitelli, 1, 06123 Perugia, Italy

Received  September 2011 Published  May 2013

The paper deals with local well--posedness, global existence and blow--up results for reaction--diffusion equations coupled with nonlinear dynamical boundary conditions. The typical problem studied is \[\begin{cases} u_{t}-\Delta u=|u|^{p-2} u        in   (0,\infty)\times\Omega,\\ u=0                    on     [0,\infty) \times \Gamma_{0},\\ \frac{\partial u}{\partial\nu} = -|u_{t}|^{m-2}u_{t}       on   [0,\infty)\times\Gamma_{1},\\ u(0,x)=u_{0}(x)         in   \Omega \end{cases}\] where $\Omega$ is a bounded open regular domain of $\mathbb{R}^{n}$ ($n\geq 1$), $\partial\Omega=\Gamma_0\cup\Gamma_1$, $2\le p\le 1+2^*/2$, $m>1$ and $u_0\in H^1(\Omega)$, ${u_0}_{|\Gamma_0}=0$. After showing local well--posedness in the Hadamard sense we give global existence and blow--up results when $\Gamma_0$ has positive surface measure. Moreover we discuss the generalization of the above mentioned results to more general problems where the terms $|u|^{p-2}u$ and $|u_{t}|^{m-2}u_{t}$ are respectively replaced by $f\left(x,u\right)$ and $Q(t,x,u_t)$ under suitable assumptions on them.
Keywords: Heat equation, reaction diffusion equations, blow--up, Hadamard well--posedness., dynamical boundary conditions.
Mathematics Subject Classification: Primary: 35K20, 35B44; Secondary: 35K57, 35Q79, 35K5.
Citation: Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015
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show all references

References:
[1]

Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, 65, 1975.  Google Scholar

[2]

J. Differential Equations, 72 (1988), 201-269. doi: 10.1016/0022-0396(88)90156-8.  Google Scholar

[3]

C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.  Google Scholar

[4]

Nonlinear Anal., 73 (2010), 1952-1965. doi: 10.1016/j.na.2010.05.024.  Google Scholar

[5]

Commun. Pure Appl. Anal., 9 (2010), 1161-1188. doi: 10.3934/cpaa.2010.9.1161.  Google Scholar

[6]

Arch. Ration. Mech. Anal., 196 (2010), 489-516. doi: 10.1007/s00205-009-0241-x.  Google Scholar

[7]

I. Bejenaru, J. I. Díaz and I. I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic boundary conditions,, Electron. J. Differential Equations, 2001 ().   Google Scholar

[8]

Discrete Contin. Dyn. Syst., 22 (2008), 835-860. doi: 10.3934/dcds.2008.22.835.  Google Scholar

[9]

J. Differential Equations, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009.  Google Scholar

[10]

Universitext, Springer, New York, 2011.  Google Scholar

[11]

H. Brezis and T. Cazenave, Unpublished, Book., ().   Google Scholar

[12]

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[13]

J. Differential Equations, 236 (2007), 407-459. doi: 10.1016/j.jde.2007.02.004.  Google Scholar

[14]

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[15]

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[16]

McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar

[17]

Japan J. Indust. Appl. Math., 9 (1992), 181-203. doi: 10.1007/BF03167565.  Google Scholar

[18]

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[19]

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[21]

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[22]

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[23]

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[43]

Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

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Nonlinear Anal., 50 (2002), Ser. A: Theory Methods, 223-250. doi: 10.1016/S0362-546X(01)00748-9.  Google Scholar

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Arch. Rational Mech. Anal., 45 (1972), 294-320.  Google Scholar

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J. Differential Equations, 193 (2003), 212-238. doi: 10.1016/S0022-0396(03)00128-1.  Google Scholar

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Appl. Anal., 87 (2008), 699-707. doi: 10.1080/00036810802189662.  Google Scholar

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Int. J. Pure Appl. Math., 48 (2008), 193-202.  Google Scholar

[49]

J. Math. Anal. Appl., 354 (2009), 394-396. doi: 10.1016/j.jmaa.2009.01.010.  Google Scholar

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J. Differential Equations, 150 (1998), 203-214. doi: 10.1006/jdeq.1998.3477.  Google Scholar

[51]

Discrete Contin. Dyn. Syst., 18 (2007), 15-38. doi: 10.3934/dcds.2007.18.15.  Google Scholar

[52]

Differential Integral Equations, 16 (2003), 13-50.  Google Scholar

[53]

Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[54]

Pacific J. Math., 19 (1966), 543-551. doi: 10.2140/pjm.1966.19.543.  Google Scholar

[55]

Applied Mathematical Sciences, 117, Springer-Verlag, New York, 1997, Nonlinear Equations, Corrected Reprint of the 1996 Original.  Google Scholar

[56]

Soviet Math. Dokl., 1 (1960), 132-133.  Google Scholar

[57]

Arch. Rational Mech. Anal., 149 (1999), 155-182. doi: 10.1007/s002050050171.  Google Scholar

[58]

Rend. Istit. Mat. Univ. Trieste, 31 (2000), 245-275, Workshop on Blow-up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999).  Google Scholar

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Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1-33. doi: 10.1017/S0308210500003838.  Google Scholar

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Comm. Partial Differential Equations, 28 (2003), 223-247. doi: 10.1081/PDE-120019380.  Google Scholar

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Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 1031-1041.  Google Scholar

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Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989, Sobolev spaces and functions of bounded variation. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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