November  2012, 11(6): 2239-2260. doi: 10.3934/cpaa.2012.11.2239

Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates

1. 

Center of Smart Interfaces, Technical University Darmstadt, Petersenstr. 32, 64287 Darmstadt

2. 

Duisburg-Essen University, Faculty of Mathematics, Universitatsstrae 2, 45141 Essen, Germany

Received  March 2011 Revised  July 2011 Published  April 2012

We consider reaction-diffusion systems with merely measurable reaction terms to cover the possibility of discontinuities. Solutions of such problems are defined as solutions to appropriate differential inclusions which, in an abstract form, lead to evolution inclusions of the form

$u' \in - A u + F(t,u)$ on $[0,T], u(0)=u_{0},$

where $A$ is $m$-accretive and $F$ is of upper semicontinuous type. While such problems, in general, can exhibit non-existence of solutions, the present paper shows that especially for $m$-completely accretive $A$, and under reasonable assumptions on $F$, mild solutions do exist.

Citation: Dieter Bothe, Petra Wittbold. Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2239-2260. doi: 10.3934/cpaa.2012.11.2239
References:
[1]

F. Andreu, N. Igbida, J. M. Mazon and J. Toledo, $L^1$ existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 24 (2007), 61. doi: 10.1016/j.anihpc.2005.09.009. Google Scholar

[2]

H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces,, Israel J. Math., 12 (1972), 373. doi: 10.1007/BF02764629. Google Scholar

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff, (1976). Google Scholar

[4]

Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations,, Ann. Sc. Norm. Sup. Pisa, 22 (1995), 241. Google Scholar

[5]

Ph. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations,, in, (1981), 23. Google Scholar

[6]

Ph. Bénilan and M. G. Crandall, Completely accretive operators,, in, 135 (1991), 41. Google Scholar

[7]

Ph. Bénilan, M. G. Crandall and A. Pazy, "Nonlinear Evolution Equations in Banach Spaces,", Preprint book., (). Google Scholar

[8]

D. Bothe, Minimal solutions of multivalued differential equations,, Diff. and Integral Eqs., 4 (1991), 445. Google Scholar

[9]

D. Bothe, Flow invariance for perturbed nonlinear evolution equations,, Abstract and Applied Analysis, 1 (1996), 417. doi: 10.1155/S1085337596000231. Google Scholar

[10]

D. Bothe, Reaction-diffusion systems with discontinuities. A viability approach,, in, 30 (1997), 677. doi: 10.1016/S0362-546X(97)00247-2. Google Scholar

[11]

D. Bothe, Multivalued perturbations of $m$-accretive differential inclusions,, Israel J. Math., 108 (1998), 109. doi: 10.1007/BF02783044. Google Scholar

[12]

D. Bothe, Periodic solutions of non-smooth friction oscillators,, Z. Angew. Math. Phys., 50 (1999), 779. doi: 10.1007/s000330050178. Google Scholar

[13]

D. Bothe, "Nonlinear Evolutions in Banach Spaces - Existence and Qualitative Theory with Applications to Reaction-Diffusion Systems,", Habilitation thesis, (1999). Google Scholar

[14]

D. Bothe, Nonlinear evolutions with Carathéodory forcing,, J. Evol. Eqs., 3 (2003), 375. doi: 10.1007/s00028-003-0099-5. Google Scholar

[15]

D. Bothe, Flow invariance for nonlinear accretive evolutions under range conditions,, J. Evol. Eqs., 5 (2005), 227. doi: 10.1007/s00028-005-0185-z. Google Scholar

[16]

K. Deimling, "Multivalued Differential Equations,", De Gruyter 1992., (1992). Google Scholar

[17]

K. Deimling, G. Hetzer and W. Shen, Almost periodicity enforced by Coulomb friction,, Advances in Diff. Eqs., 1 (1996), 265. Google Scholar

[18]

J. I. Diaz, Diffusive energy balance models in climatology,, in, (2002), 297. doi: 10.1016/S0168-2024(02)80015-7. Google Scholar

[19]

J. I. Diaz and I. I. Vrabie, Existence for reaction diffusion systems. A compactness method approach,, J. Math. Anal. Appl., 188 (1994), 521. Google Scholar

[20]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^1(\mu,X)$,, in, 118 (1993), 447. doi: 10.1090/S0002-9939-1993-1132408-x. Google Scholar

[21]

G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics,", Springer, (1976). doi: 10.1007/978-3-642-66165-5. Google Scholar

[22]

H. O. Fattorini, Infinite dimensional optimization and control theory,, in, (1999). Google Scholar

[23]

E. Feireisl and J. Norbury, Some existence, uniqueness and non-uniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities,, in, 119A (1991), 1. Google Scholar

[24]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides,", Kluwer, (1988). doi: 10.1016/0378-4754(89)90171-7. Google Scholar

[25]

L. Górniewicz, A. Granas and W. Kryszewski, Sur la méthode de l'homotopie dans la théorie des point fixes pour les applications multivoques. Partie 2: L'indice dans les ANRs compacts,, C. R. Acad. Sci. Paris, 308 (1989), 449. Google Scholar

[26]

V. G. Jakubowski and P. Wittbold, Regularity of solutions of nonlinear Volterra equations,, J. Evol. Equ., 3 (2003), 303. doi: 10.1007/s00028-003-0096-9. Google Scholar

[27]

R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems,, in, 185 (1992), 363. doi: 10.1016/S0076-5392(08)62804-0. Google Scholar

[28]

I. Miyadera, "Nonlinear Semigroups,", Translations of Math. Monographs 109, (1992). Google Scholar

[29]

J. Norbury and A. M. Stuart, A model for porous medium combustion,, Quart. J. Mech. Appl. Math., 42 (1987), 159. doi: 10.1093/qjmam/42.1.159. Google Scholar

[30]

M. Pierre, Un théorème général de génération de semi-groupes non linéaires,, Israel J. Math., 23 (1976), 189. Google Scholar

[31]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: a survey,, Milan J. Math., 78 (2010), 417. doi: 10.1007/s00032-010-0133-4. Google Scholar

[32]

T. Rzezuchowski, Scorza-Dragoni type theorem for upper semicontinuous multivalued functions,, Bull. Acad. Polon. Sci., 28 (1980), 61. Google Scholar

[33]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Grundlehren math. Wissenschaften {\bf258}, (1983). Google Scholar

[34]

A. M. Stuart, The mathematics of porous medium combustion,, in, (1988), 295. Google Scholar

[35]

A. A. Tolstonogov and Y. I. Umanskii, On solutions of evolution inclusions II,, Sib. Math. J., 33 (1992), 693. doi: 10.1007/BF00971135. Google Scholar

[36]

M. Valencia, On invariant regions and asymptotic bounds for semilinear partial differential equations,, Nonlinear Analysis, 14 (1990), 217. doi: 10.1016/0362-546X(90)90030-K. Google Scholar

[37]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994). Google Scholar

[38]

I. I. Vrabie, "Compactness Methods for Nonlinear Evolutions," $2^{nd}$ edition,, Pitman, (1995). Google Scholar

show all references

References:
[1]

F. Andreu, N. Igbida, J. M. Mazon and J. Toledo, $L^1$ existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 24 (2007), 61. doi: 10.1016/j.anihpc.2005.09.009. Google Scholar

[2]

H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces,, Israel J. Math., 12 (1972), 373. doi: 10.1007/BF02764629. Google Scholar

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff, (1976). Google Scholar

[4]

Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations,, Ann. Sc. Norm. Sup. Pisa, 22 (1995), 241. Google Scholar

[5]

Ph. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations,, in, (1981), 23. Google Scholar

[6]

Ph. Bénilan and M. G. Crandall, Completely accretive operators,, in, 135 (1991), 41. Google Scholar

[7]

Ph. Bénilan, M. G. Crandall and A. Pazy, "Nonlinear Evolution Equations in Banach Spaces,", Preprint book., (). Google Scholar

[8]

D. Bothe, Minimal solutions of multivalued differential equations,, Diff. and Integral Eqs., 4 (1991), 445. Google Scholar

[9]

D. Bothe, Flow invariance for perturbed nonlinear evolution equations,, Abstract and Applied Analysis, 1 (1996), 417. doi: 10.1155/S1085337596000231. Google Scholar

[10]

D. Bothe, Reaction-diffusion systems with discontinuities. A viability approach,, in, 30 (1997), 677. doi: 10.1016/S0362-546X(97)00247-2. Google Scholar

[11]

D. Bothe, Multivalued perturbations of $m$-accretive differential inclusions,, Israel J. Math., 108 (1998), 109. doi: 10.1007/BF02783044. Google Scholar

[12]

D. Bothe, Periodic solutions of non-smooth friction oscillators,, Z. Angew. Math. Phys., 50 (1999), 779. doi: 10.1007/s000330050178. Google Scholar

[13]

D. Bothe, "Nonlinear Evolutions in Banach Spaces - Existence and Qualitative Theory with Applications to Reaction-Diffusion Systems,", Habilitation thesis, (1999). Google Scholar

[14]

D. Bothe, Nonlinear evolutions with Carathéodory forcing,, J. Evol. Eqs., 3 (2003), 375. doi: 10.1007/s00028-003-0099-5. Google Scholar

[15]

D. Bothe, Flow invariance for nonlinear accretive evolutions under range conditions,, J. Evol. Eqs., 5 (2005), 227. doi: 10.1007/s00028-005-0185-z. Google Scholar

[16]

K. Deimling, "Multivalued Differential Equations,", De Gruyter 1992., (1992). Google Scholar

[17]

K. Deimling, G. Hetzer and W. Shen, Almost periodicity enforced by Coulomb friction,, Advances in Diff. Eqs., 1 (1996), 265. Google Scholar

[18]

J. I. Diaz, Diffusive energy balance models in climatology,, in, (2002), 297. doi: 10.1016/S0168-2024(02)80015-7. Google Scholar

[19]

J. I. Diaz and I. I. Vrabie, Existence for reaction diffusion systems. A compactness method approach,, J. Math. Anal. Appl., 188 (1994), 521. Google Scholar

[20]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^1(\mu,X)$,, in, 118 (1993), 447. doi: 10.1090/S0002-9939-1993-1132408-x. Google Scholar

[21]

G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics,", Springer, (1976). doi: 10.1007/978-3-642-66165-5. Google Scholar

[22]

H. O. Fattorini, Infinite dimensional optimization and control theory,, in, (1999). Google Scholar

[23]

E. Feireisl and J. Norbury, Some existence, uniqueness and non-uniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities,, in, 119A (1991), 1. Google Scholar

[24]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides,", Kluwer, (1988). doi: 10.1016/0378-4754(89)90171-7. Google Scholar

[25]

L. Górniewicz, A. Granas and W. Kryszewski, Sur la méthode de l'homotopie dans la théorie des point fixes pour les applications multivoques. Partie 2: L'indice dans les ANRs compacts,, C. R. Acad. Sci. Paris, 308 (1989), 449. Google Scholar

[26]

V. G. Jakubowski and P. Wittbold, Regularity of solutions of nonlinear Volterra equations,, J. Evol. Equ., 3 (2003), 303. doi: 10.1007/s00028-003-0096-9. Google Scholar

[27]

R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems,, in, 185 (1992), 363. doi: 10.1016/S0076-5392(08)62804-0. Google Scholar

[28]

I. Miyadera, "Nonlinear Semigroups,", Translations of Math. Monographs 109, (1992). Google Scholar

[29]

J. Norbury and A. M. Stuart, A model for porous medium combustion,, Quart. J. Mech. Appl. Math., 42 (1987), 159. doi: 10.1093/qjmam/42.1.159. Google Scholar

[30]

M. Pierre, Un théorème général de génération de semi-groupes non linéaires,, Israel J. Math., 23 (1976), 189. Google Scholar

[31]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: a survey,, Milan J. Math., 78 (2010), 417. doi: 10.1007/s00032-010-0133-4. Google Scholar

[32]

T. Rzezuchowski, Scorza-Dragoni type theorem for upper semicontinuous multivalued functions,, Bull. Acad. Polon. Sci., 28 (1980), 61. Google Scholar

[33]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Grundlehren math. Wissenschaften {\bf258}, (1983). Google Scholar

[34]

A. M. Stuart, The mathematics of porous medium combustion,, in, (1988), 295. Google Scholar

[35]

A. A. Tolstonogov and Y. I. Umanskii, On solutions of evolution inclusions II,, Sib. Math. J., 33 (1992), 693. doi: 10.1007/BF00971135. Google Scholar

[36]

M. Valencia, On invariant regions and asymptotic bounds for semilinear partial differential equations,, Nonlinear Analysis, 14 (1990), 217. doi: 10.1016/0362-546X(90)90030-K. Google Scholar

[37]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994). Google Scholar

[38]

I. I. Vrabie, "Compactness Methods for Nonlinear Evolutions," $2^{nd}$ edition,, Pitman, (1995). Google Scholar

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