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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 - A, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015
References:
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H. Amann, Parabolic evolution equations and nonlinear boundary conditions,, J. Differential Equations, 72 (1988), 201. doi: 10.1016/0022-0396(88)90156-8. Google Scholar

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J.-P. Aubin, Un théorème de compacité,, C. R. Acad. Sci. Paris, 256 (1963), 5042. Google Scholar

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G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems,, Arch. Ration. Mech. Anal., 196 (2010), 489. doi: 10.1007/s00205-009-0241-x. Google Scholar

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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]

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. doi: 10.3934/dcds.2008.22.835. Google Scholar

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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. doi: 10.1016/j.jde.2004.04.011. Google Scholar

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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. doi: 10.1016/j.camwa.2010.05.015. Google Scholar

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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. Google Scholar

[26]

G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation,, J. Math. Anal. Appl., 333 (2007), 839. doi: 10.1016/j.jmaa.2006.11.050. Google Scholar

[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. doi: 10.1017/S0308210500021946. Google Scholar

[28]

T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43. doi: 10.1017/S0308210500023945. Google Scholar

[29]

K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion,, J. Differential Equations, 212 (2005), 114. doi: 10.1016/j.jde.2004.10.021. Google Scholar

[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. doi: 10.1016/j.anihpc.2006.12.002. Google Scholar

[31]

M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type,, Hokkaido Math. J., 21 (1992), 221. Google Scholar

[32]

I. Lasiecka, Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions,, J. Differential Equations, 75 (1988), 53. doi: 10.1016/0022-0396(88)90129-5. Google Scholar

[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. Google Scholar

[34]

______, The role of critical exponents in blowup theorems,, SIAM Rev., 32 (1990), 262. doi: 10.1137/1032046. Google Scholar

[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. doi: 10.1006/jdeq.1997.3362. Google Scholar

[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. doi: 10.1016/0022-0396(74)90018-7. Google Scholar

[37]

______, Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints,, Proc. Amer. Math. Soc. 46 (1974), 46 (1974), 277. Google Scholar

[38]

H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation,, Arch. Rational Mech. Anal., 137 (1997), 341. doi: 10.1007/s002050050032. Google Scholar

[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. doi: 10.1002/mma.1670090111. Google Scholar

[40]

______, A potential well theory for the wave equation with a nonlinear boundary condition,, J. Reine Angew. Math., 374 (1987), 1. Google Scholar

[41]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1,", Travaux et Recherches Mathématiques, (1968). Google Scholar

[42]

J.-L. Lions and W. A. Strauss, Some non-linear evolution equations,, Bull. Soc. Math. France, 93 (1965), 43. Google Scholar

[43]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Progress in Nonlinear Differential Equations and Their Applications, 16 (1995). doi: 10.1007/978-3-0348-9234-6. Google Scholar

[44]

E. Maitre and P. Witomski, A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations,, Nonlinear Anal., 50 (2002), 223. doi: 10.1016/S0362-546X(01)00748-9. Google Scholar

[45]

M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces,, Arch. Rational Mech. Anal., 45 (1972), 294. Google Scholar

[46]

N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition,, J. Differential Equations, 193 (2003), 212. doi: 10.1016/S0022-0396(03)00128-1. Google Scholar

[47]

L. E. Payne and P. W. Schaefer, Blow-up in parabolic problems under Robin boundary conditions,, Appl. Anal., 87 (2008), 699. doi: 10.1080/00036810802189662. Google Scholar

[48]

_______, Blow-up phenomena for some nonlinear parabolic systems,, Int. J. Pure Appl. Math., 48 (2008), 193. Google Scholar

[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. doi: 10.1016/j.jmaa.2009.01.010. Google Scholar

[50]

P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy,, J. Differential Equations, 150 (1998), 203. doi: 10.1006/jdeq.1998.3477. Google Scholar

[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. doi: 10.3934/dcds.2007.18.15. Google Scholar

[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. Google Scholar

[53]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[54]

W. A. Strauss, On continuity of functions with values in various Banach spaces,, Pacific J. Math., 19 (1966), 543. doi: 10.2140/pjm.1966.19.543. Google Scholar

[55]

M. E. Taylor, "Partial Differential Equations. III,", Applied Mathematical Sciences, 117 (1997). Google Scholar

[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. Google Scholar

[57]

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation and application,, Arch. Rational Mech. Anal., 149 (1999), 155. doi: 10.1007/s002050050171. Google Scholar

[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. Google Scholar

[59]

______, Global existence for the wave equation with nonlinear boundary damping and source terms,, J. Differential Equations, 186 (2002), 259. doi: 10.1016/S0022-0396(02)00023-2. Google Scholar

[60]

______, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375. doi: 10.1017/S0017089502030045. Google Scholar

[61]

______, Global existence for the heat equation with nonlinear dynamical boundary condition,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1. doi: 10.1017/S0308210500003838. Google Scholar

[62]

_______, On the Laplace equation with non-linear dynamical boundary conditions,, Proc. London Math. Soc. (3), 93 (2006), 418. doi: 10.1112/S0024611506015875. Google Scholar

[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. doi: 10.1081/PDE-120019380. Google Scholar

[64]

______, "Blow Up for Some Nonlinear Parabolic Problems with Convection Under Dynamical Boundary Conditions,", Discrete Contin. Dyn. Syst., (2007), 1031. Google Scholar

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W. P. Ziemer, "Weakly Differentiable Functions,", Graduate Texts in Mathematics, 120, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,", Academic Press [A subsidiary of Harcourt Brace Jovanovich, 65 (1975). Google Scholar

[2]

H. Amann, Parabolic evolution equations and nonlinear boundary conditions,, J. Differential Equations, 72 (1988), 201. doi: 10.1016/0022-0396(88)90156-8. Google Scholar

[3]

J.-P. Aubin, Un théorème de compacité,, C. R. Acad. Sci. Paris, 256 (1963), 5042. Google Scholar

[4]

G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions,, Nonlinear Anal., 73 (2010), 1952. doi: 10.1016/j.na.2010.05.024. Google Scholar

[5]

______, Kirchhoff systems with nonlinear source and boundary damping terms,, Commun. Pure Appl. Anal., 9 (2010), 1161. doi: 10.3934/cpaa.2010.9.1161. Google Scholar

[6]

G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems,, Arch. Ration. Mech. Anal., 196 (2010), 489. 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]

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. doi: 10.3934/dcds.2008.22.835. Google Scholar

[9]

______, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654. doi: 10.1016/j.jde.2010.03.009. Google Scholar

[10]

H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Universitext, (2011). Google Scholar

[11]

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

[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. Google Scholar

[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. doi: 10.1016/j.jde.2007.02.004. Google Scholar

[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. doi: 10.1016/j.jde.2004.04.011. Google Scholar

[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. doi: 10.1081/PDE-120016132. Google Scholar

[16]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill Book Company, (1955). Google Scholar

[17]

P. Colli, On some doubly nonlinear evolution equations in Banach spaces,, Japan J. Indust. Appl. Math., 9 (1992), 181. doi: 10.1007/BF03167565. Google Scholar

[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. doi: 10.1016/j.camwa.2010.05.015. Google Scholar

[19]

J. Escher, Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions,, Math. Ann., 284 (1989), 285. doi: 10.1007/BF01442877. Google Scholar

[20]

______, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309. doi: 10.1080/03605309308820976. Google Scholar

[21]

______, On the qualitative behaviour of some semilinear parabolic problems,, Differential Integral Equations, 8 (1995), 247. Google Scholar

[22]

Z.-H. Fan and C.-K. Zhong, Attractors for parabolic equations with dynamic boundary conditions,, Nonlinear Anal., 68 (2008), 1723. doi: 10.1016/j.na.2007.01.005. Google Scholar

[23]

V. A. Galaktionov and J. L. Vázquez, The problem of blow-up in nonlinear parabolic equations,, Discrete Contin. Dyn. Syst., 8 (2002), 399. doi: 10.3934/dcds.2002.8.399. Google Scholar

[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. doi: 10.1006/jdeq.1994.1051. Google Scholar

[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. Google Scholar

[26]

G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation,, J. Math. Anal. Appl., 333 (2007), 839. doi: 10.1016/j.jmaa.2006.11.050. Google Scholar

[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. doi: 10.1017/S0308210500021946. Google Scholar

[28]

T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43. doi: 10.1017/S0308210500023945. Google Scholar

[29]

K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion,, J. Differential Equations, 212 (2005), 114. doi: 10.1016/j.jde.2004.10.021. Google Scholar

[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. doi: 10.1016/j.anihpc.2006.12.002. Google Scholar

[31]

M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type,, Hokkaido Math. J., 21 (1992), 221. Google Scholar

[32]

I. Lasiecka, Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions,, J. Differential Equations, 75 (1988), 53. doi: 10.1016/0022-0396(88)90129-5. Google Scholar

[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. Google Scholar

[34]

______, The role of critical exponents in blowup theorems,, SIAM Rev., 32 (1990), 262. doi: 10.1137/1032046. Google Scholar

[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. doi: 10.1006/jdeq.1997.3362. Google Scholar

[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. doi: 10.1016/0022-0396(74)90018-7. Google Scholar

[37]

______, Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints,, Proc. Amer. Math. Soc. 46 (1974), 46 (1974), 277. Google Scholar

[38]

H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation,, Arch. Rational Mech. Anal., 137 (1997), 341. doi: 10.1007/s002050050032. Google Scholar

[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. doi: 10.1002/mma.1670090111. Google Scholar

[40]

______, A potential well theory for the wave equation with a nonlinear boundary condition,, J. Reine Angew. Math., 374 (1987), 1. Google Scholar

[41]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1,", Travaux et Recherches Mathématiques, (1968). Google Scholar

[42]

J.-L. Lions and W. A. Strauss, Some non-linear evolution equations,, Bull. Soc. Math. France, 93 (1965), 43. Google Scholar

[43]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Progress in Nonlinear Differential Equations and Their Applications, 16 (1995). doi: 10.1007/978-3-0348-9234-6. Google Scholar

[44]

E. Maitre and P. Witomski, A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations,, Nonlinear Anal., 50 (2002), 223. doi: 10.1016/S0362-546X(01)00748-9. Google Scholar

[45]

M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces,, Arch. Rational Mech. Anal., 45 (1972), 294. Google Scholar

[46]

N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition,, J. Differential Equations, 193 (2003), 212. doi: 10.1016/S0022-0396(03)00128-1. Google Scholar

[47]

L. E. Payne and P. W. Schaefer, Blow-up in parabolic problems under Robin boundary conditions,, Appl. Anal., 87 (2008), 699. doi: 10.1080/00036810802189662. Google Scholar

[48]

_______, Blow-up phenomena for some nonlinear parabolic systems,, Int. J. Pure Appl. Math., 48 (2008), 193. Google Scholar

[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. doi: 10.1016/j.jmaa.2009.01.010. Google Scholar

[50]

P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy,, J. Differential Equations, 150 (1998), 203. doi: 10.1006/jdeq.1998.3477. Google Scholar

[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. doi: 10.3934/dcds.2007.18.15. Google Scholar

[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. Google Scholar

[53]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[54]

W. A. Strauss, On continuity of functions with values in various Banach spaces,, Pacific J. Math., 19 (1966), 543. doi: 10.2140/pjm.1966.19.543. Google Scholar

[55]

M. E. Taylor, "Partial Differential Equations. III,", Applied Mathematical Sciences, 117 (1997). Google Scholar

[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. Google Scholar

[57]

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation and application,, Arch. Rational Mech. Anal., 149 (1999), 155. doi: 10.1007/s002050050171. Google Scholar

[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. Google Scholar

[59]

______, Global existence for the wave equation with nonlinear boundary damping and source terms,, J. Differential Equations, 186 (2002), 259. doi: 10.1016/S0022-0396(02)00023-2. Google Scholar

[60]

______, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375. doi: 10.1017/S0017089502030045. Google Scholar

[61]

______, Global existence for the heat equation with nonlinear dynamical boundary condition,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1. doi: 10.1017/S0308210500003838. Google Scholar

[62]

_______, On the Laplace equation with non-linear dynamical boundary conditions,, Proc. London Math. Soc. (3), 93 (2006), 418. doi: 10.1112/S0024611506015875. Google Scholar

[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. doi: 10.1081/PDE-120019380. Google Scholar

[64]

______, "Blow Up for Some Nonlinear Parabolic Problems with Convection Under Dynamical Boundary Conditions,", Discrete Contin. Dyn. Syst., (2007), 1031. Google Scholar

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