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

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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

Abstract
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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 and Continuous Dynamical Systems, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015

References:

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[2]	H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269. doi: 10.1016/0022-0396(88)90156-8.
[3]	J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.
[4]	G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73 (2010), 1952-1965. doi: 10.1016/j.na.2010.05.024.
[5]	______, Kirchhoff systems with nonlinear source and boundary damping terms, Commun. Pure Appl. Anal., 9 (2010), 1161-1188. doi: 10.3934/cpaa.2010.9.1161.
[6]	G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516. doi: 10.1007/s00205-009-0241-x.
[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 ().
[8]	L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860. doi: 10.3934/dcds.2008.22.835.
[9]	______, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009.
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[15]	I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132.
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[26]	G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation, J. Math. Anal. Appl., 333 (2007), 839-862. doi: 10.1016/j.jmaa.2006.11.050.
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[28]	T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43-60. doi: 10.1017/S0308210500023945.
[29]	K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations, 212 (2005), 114-128. doi: 10.1016/j.jde.2004.10.021.
[30]	M. Jazar and R. Kiwan, Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 215-218. doi: 10.1016/j.anihpc.2006.12.002.
[31]	M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type, Hokkaido Math. J., 21 (1992), 221-229.
[32]	I. Lasiecka, Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions, J. Differential Equations, 75 (1988), 53-87. doi: 10.1016/0022-0396(88)90129-5.
[33]	H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+\mathcalF(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.
[34]	______, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288. doi: 10.1137/1032046.
[35]	H. A. Levine, S. R. Park and J. Serrin, Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type, J. Differential Equations, 142 (1998), 212-229. doi: 10.1006/jdeq.1997.3362.
[36]	H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations, 16 (1974), 319-334. doi: 10.1016/0022-0396(74)90018-7.
[37]	______, Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints, Proc. Amer. Math. Soc. 46 (1974), 277-284.
[38]	H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361. doi: 10.1007/s002050050032.
[39]	H. A. Levine and R. A. Smith, A potential well theory for the heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci., 9 (1987), 127-136. doi: 10.1002/mma.1670090111.
[40]	______, A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math., 374 (1987), 1-23.
[41]	J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1," Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.
[42]	J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.
[43]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.
[44]	E. Maitre and P. Witomski, A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations, Nonlinear Anal., 50 (2002), Ser. A: Theory Methods, 223-250. doi: 10.1016/S0362-546X(01)00748-9.
[45]	M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal., 45 (1972), 294-320.
[46]	N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition, J. Differential Equations, 193 (2003), 212-238. doi: 10.1016/S0022-0396(03)00128-1.
[47]	L. E. Payne and P. W. Schaefer, Blow-up in parabolic problems under Robin boundary conditions, Appl. Anal., 87 (2008), 699-707. doi: 10.1080/00036810802189662.
[48]	_______, Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202.
[49]	L. E. Payne and J. C. Song, Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl., 354 (2009), 394-396. doi: 10.1016/j.jmaa.2009.01.010.
[50]	P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214. doi: 10.1006/jdeq.1998.3477.
[51]	G. Schimperna, A. Segatti and U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations, Discrete Contin. Dyn. Syst., 18 (2007), 15-38. doi: 10.3934/dcds.2007.18.15.
[52]	J. Serrin, G. Todorova and E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Differential Integral Equations, 16 (2003), 13-50.
[53]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.
[54]	W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551. doi: 10.2140/pjm.1966.19.543.
[55]	M. E. Taylor, "Partial Differential Equations. III," Applied Mathematical Sciences, 117, Springer-Verlag, New York, 1997, Nonlinear Equations, Corrected Reprint of the 1996 Original.
[56]	S. V. Uspenskiĭ, An imbedding theorem for S. L. Sobolev's classes of fractional order $W_{p^r}$, Soviet Math. Dokl., 1 (1960), 132-133.
[57]	E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation and application, Arch. Rational Mech. Anal., 149 (1999), 155-182. doi: 10.1007/s002050050171.
[58]	______, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, 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).
[59]	______, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298. doi: 10.1016/S0022-0396(02)00023-2.
[60]	______, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395. doi: 10.1017/S0017089502030045.
[61]	______, Global existence for the heat equation with nonlinear dynamical boundary condition, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1-33. doi: 10.1017/S0308210500003838.
[62]	_______, On the Laplace equation with non-linear dynamical boundary conditions, Proc. London Math. Soc. (3), 93 (2006), 418-446. doi: 10.1112/S0024611506015875.
[63]	J. von Below and G. Pincet Mailly, Blow up for reaction diffusion equations under dynamical boundary conditions, Comm. Partial Differential Equations, 28 (2003), 223-247. doi: 10.1081/PDE-120019380.
[64]	______, "Blow Up for Some Nonlinear Parabolic Problems with Convection Under Dynamical Boundary Conditions," Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 1031-1041.
[65]	W. P. Ziemer, "Weakly Differentiable Functions," 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.

show all references

References:

[1]	R. A. Adams, "Sobolev Spaces," Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, 65, 1975.
[2]	H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269. doi: 10.1016/0022-0396(88)90156-8.
[3]	J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.
[4]	G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73 (2010), 1952-1965. doi: 10.1016/j.na.2010.05.024.
[5]	______, Kirchhoff systems with nonlinear source and boundary damping terms, Commun. Pure Appl. Anal., 9 (2010), 1161-1188. doi: 10.3934/cpaa.2010.9.1161.
[6]	G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516. doi: 10.1007/s00205-009-0241-x.
[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 ().
[8]	L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860. doi: 10.3934/dcds.2008.22.835.
[9]	______, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009.
[10]	H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011.
[11]	H. Brezis and T. Cazenave, Unpublished, Book., ().
[12]	H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.
[13]	M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459. doi: 10.1016/j.jde.2007.02.004.
[14]	M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158. doi: 10.1016/j.jde.2004.04.011.
[15]	I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132.
[16]	E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
[17]	P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math., 9 (1992), 181-203. doi: 10.1007/BF03167565.
[18]	J. Ding and B.-Z. Guo, Blow-up and global existence for nonlinear parabolic equations with Neumann boundary conditions, Comput. Math. Appl., 60 (2010), 670-679. doi: 10.1016/j.camwa.2010.05.015.
[19]	J. Escher, Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions, Math. Ann., 284 (1989), 285-305. doi: 10.1007/BF01442877.
[20]	______, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976.
[21]	______, On the qualitative behaviour of some semilinear parabolic problems, Differential Integral Equations, 8 (1995), 247-267.
[22]	Z.-H. Fan and C.-K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal., 68 (2008), 1723-1732. doi: 10.1016/j.na.2007.01.005.
[23]	V. A. Galaktionov and J. L. Vázquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst., 8 (2002), 399-433, Current Developments in Partial Differential Equations (Temuco, 1999). doi: 10.3934/dcds.2002.8.399.
[24]	V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308. doi: 10.1006/jdeq.1994.1051.
[25]	S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations, 13 (2008), 1051-1074.
[26]	G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation, J. Math. Anal. Appl., 333 (2007), 839-862. doi: 10.1016/j.jmaa.2006.11.050.
[27]	M. Grobbelaar-van Dalsen, Semilinear evolution equations and fractional powers of a closed pair of operators, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), 101-115. doi: 10.1017/S0308210500021946.
[28]	T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43-60. doi: 10.1017/S0308210500023945.
[29]	K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations, 212 (2005), 114-128. doi: 10.1016/j.jde.2004.10.021.
[30]	M. Jazar and R. Kiwan, Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 215-218. doi: 10.1016/j.anihpc.2006.12.002.
[31]	M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type, Hokkaido Math. J., 21 (1992), 221-229.
[32]	I. Lasiecka, Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions, J. Differential Equations, 75 (1988), 53-87. doi: 10.1016/0022-0396(88)90129-5.
[33]	H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+\mathcalF(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.
[34]	______, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288. doi: 10.1137/1032046.
[35]	H. A. Levine, S. R. Park and J. Serrin, Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type, J. Differential Equations, 142 (1998), 212-229. doi: 10.1006/jdeq.1997.3362.
[36]	H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations, 16 (1974), 319-334. doi: 10.1016/0022-0396(74)90018-7.
[37]	______, Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints, Proc. Amer. Math. Soc. 46 (1974), 277-284.
[38]	H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361. doi: 10.1007/s002050050032.
[39]	H. A. Levine and R. A. Smith, A potential well theory for the heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci., 9 (1987), 127-136. doi: 10.1002/mma.1670090111.
[40]	______, A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math., 374 (1987), 1-23.
[41]	J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1," Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.
[42]	J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.
[43]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.
[44]	E. Maitre and P. Witomski, A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations, Nonlinear Anal., 50 (2002), Ser. A: Theory Methods, 223-250. doi: 10.1016/S0362-546X(01)00748-9.
[45]	M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal., 45 (1972), 294-320.
[46]	N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition, J. Differential Equations, 193 (2003), 212-238. doi: 10.1016/S0022-0396(03)00128-1.
[47]	L. E. Payne and P. W. Schaefer, Blow-up in parabolic problems under Robin boundary conditions, Appl. Anal., 87 (2008), 699-707. doi: 10.1080/00036810802189662.
[48]	_______, Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202.
[49]	L. E. Payne and J. C. Song, Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl., 354 (2009), 394-396. doi: 10.1016/j.jmaa.2009.01.010.
[50]	P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214. doi: 10.1006/jdeq.1998.3477.
[51]	G. Schimperna, A. Segatti and U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations, Discrete Contin. Dyn. Syst., 18 (2007), 15-38. doi: 10.3934/dcds.2007.18.15.
[52]	J. Serrin, G. Todorova and E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Differential Integral Equations, 16 (2003), 13-50.
[53]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.
[54]	W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551. doi: 10.2140/pjm.1966.19.543.
[55]	M. E. Taylor, "Partial Differential Equations. III," Applied Mathematical Sciences, 117, Springer-Verlag, New York, 1997, Nonlinear Equations, Corrected Reprint of the 1996 Original.
[56]	S. V. Uspenskiĭ, An imbedding theorem for S. L. Sobolev's classes of fractional order $W_{p^r}$, Soviet Math. Dokl., 1 (1960), 132-133.
[57]	E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation and application, Arch. Rational Mech. Anal., 149 (1999), 155-182. doi: 10.1007/s002050050171.
[58]	______, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, 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).
[59]	______, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298. doi: 10.1016/S0022-0396(02)00023-2.
[60]	______, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395. doi: 10.1017/S0017089502030045.
[61]	______, Global existence for the heat equation with nonlinear dynamical boundary condition, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1-33. doi: 10.1017/S0308210500003838.
[62]	_______, On the Laplace equation with non-linear dynamical boundary conditions, Proc. London Math. Soc. (3), 93 (2006), 418-446. doi: 10.1112/S0024611506015875.
[63]	J. von Below and G. Pincet Mailly, Blow up for reaction diffusion equations under dynamical boundary conditions, Comm. Partial Differential Equations, 28 (2003), 223-247. doi: 10.1081/PDE-120019380.
[64]	______, "Blow Up for Some Nonlinear Parabolic Problems with Convection Under Dynamical Boundary Conditions," Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 1031-1041.
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