Introduction to partial differential equations

Marcelo Moreira Cavalcanti; Valéria N. Domingos Cavalcanti

doi:10.5269/bspm.81162

Marcelo Moreira Cavalcanti Universidade Estadual de Maringá, Maringá
Valéria N. Domingos Cavalcanti

Abstract

The purpose of this book is to serve students, teachers, and researchers who seek a clear and accessible introduction to modern methods in Partial Differential Equations (PDEs), particularly in regions where advanced mathematical material is not always readily available. Over the past decades, I have taught these topics in Brazil using lecture notes that grew organically from classroom experience. These notes have now been reorganized, expanded, and translated into English with a single goal in mind: to other mathematical knowledge to readers in developing countries, especially in some poor countries all over the world, where access to updated material is often limited. The topics presented here range from the classical linear theory to nonlinear analysis, monotonicity methods, compactness techniques, evolution equations, and a large variety of applied PDE models. Every chapter was written with great care for clarity and pedagogy. Technical proofs are included when necessary, but always with an effort to preserve intuition and motivation. Many of the problems reect real questions arising from physics, engineering, and contemporary mathematical research. This book is also the result of a personal commitment. I firmly believe that mathematics is a universal language - one that should not be restricted by geographical or economic barriers. The possibility of translating these materials into English and making them accessible to students around the world is, to me, an act of gratitude and service. If these pages help even a single student develop confidence and inspiration to pursue further studies, then this work will have fulfilled its purpose. I am deeply grateful to Valéria Neves Domingos Cavalcanti, whose support, friendship, and mathematical insight have shaped much of my academic life. My thanks also go to the countless students whose questions and enthusiasm gave life to these notes. It is my sincere hope that this book becomes a bridge - connecting people, ideas, and opportunities - and that it might illuminate the path for those who, like me, believe in the transformative power of education.

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References

M. Aassila, Global existence of solutions to a wave equation with damping and source terms, Differential and Integral Equations 14(11),(2001), 1301-1314.

[Al-FI] C.O. Alves and G.M. Figueiredo, Multiplicity of positive solutions for a quasilinear problem in RN via Penalization Method equation in RN, Advanced Nonlinear Studies, 5 (2005), 551 - 572.

[ALA] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim. 51(1), (2005), 61-105.

[Am-Ra] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349-381.

[A-CA-DO] M. Aassila, M. M. Cavalcanti, V. N. Domingos Cavalcanti, Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Differential Equations, 15(2) (2002), 155-180.

R.A. Adams - Sobolev Spaces, Academic Press, N.Y., 1975.

[CAL VAR(2009)] Alves, Claudianor O.; Cavalcanti, Marcelo M. On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source. Calc. Var. Partial Differential Equations 34 (2009), no. 3, 377-411.

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, 1976.

[Bar-La-Ra] V. Barbu, I. Lasiecka, A. M. Rammaha, On nonlinear wave equations with degenerate damping and source terms. Trans. Amer. Math. Soc. 357 (2005), no. 7, 2571-2611

I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical expoent and nonlinear boundary dissipation, Comm. Partial Differential Equations 27(9-10), (2002), 1901-1951.

J.A. Aubin - Approximation of Elliptic Boundary Value Problems, Wiley Interscience, N.Y., 1972.

H. Brézis - Analyse Fonctionelle (Théorie et Applications), Masson, Paris, 1973.

M.M. Cavalcanti e V.N. Domingos Cavalcanti - Iniciação à Teoria das Distribuições e aos Espaços de Sobolev, Textos Matemáticos - UEM, 2000.

-b] Cavalcanti, M. M.; Cavalcanti, V. N. Domingos; Soriano, J. A. Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients.(English summary) Electron. J. Differential Equations 1998, No. 8, 21 pp.

[CA-DO] M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and asymptotic stability for evolution problems on manifolds with damping and source terms, J. Math. Anal. Appl. 291(1), (2004), 109-127.

[CCP] M. Cavalcanti and V. 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(1) (2004), 119-158 .

-a] M. M. Cavalcanti - J. A. Soriano, On Solvability and Stability of the Wave Equation with Lower Order Terms and Boundary Damping, Revista Matemáticas Aplicadas 18(2), (1997), 61-78.

-c] Cavalcanti, Marcelo M.; Domingos Cavalcanti, Valéria N.; Martinez, Patrick. Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term. J. Differential Equations 203 (2004), no. 1, 119-158.

M. M. Cavalcanti, V. N. Domingos Cavalcanti and M. Aassila, Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. P. D. E. 15(2), (2002), 155-180.

[CA-DO-LA] M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Wellposedness and optimal decay rates for wave equation with nonlinear boundary dampingsource interaction, Journal of Differential Equations, 236(2), (2007), 407-459 .

'] G. Chen, Energy Decay Estimates and Exact Boundary Value Controllability for the Wave Equation in a Bounded Domain, J. Math. Pures et Appl. 58, (1979), 249-273.

'] G. Chen, A Note on the Boundary Stabilization of the Wave Equation, Siam J. Control and Optimization, 19(1), (1981), 106-113.

'] G. Chen, Control and Stabilization for the Wave Equation in a Bounded Domain, Siam J. Control and Optimization, 19(1), (1979), 114-122.

G. Chen and H. Wong, Asymptotic Behaviour of solutions of the one Dimensional Wave Equation with a Nonlinear Boundary Stabilizer, SIAM J. Control and Opt., 27, (1989), 758-775.

R. Dautray and J.L. Lions - Mathematical Analysis and Numerical Methods for Science and Technology, V.5, Springer-Verlag, 1983.

Y. Ebihara, M. Nakao and T. Nambu, On the xistence of global classical solution of initial boundary value proble for u′′ − ∆u − u3 = f, Pacific J. of Math. 60(1975), 63-70.

[Fig-Mi-Ruf] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in IR2 with nonlinearitis in the critical growth range, Calc. Var 3 (1995) 139-153.

[GP] V.A. Galaktionov and S.I. Pohozaev, Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Analysis, vol 53, pp 453-467, 2003.

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Diff. Equat. 109(1975), 63-70.

[Ge-To] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Diff. Equat. 109(1975), 63-70.

[Ka-Qua] L. Kaitai and Z. Quanda, Existence and nonexistence of global solutions for the equation of dislocation of crystals, J. Diff. Equat. 146, (1998), 5-21.

R. Ikehata, Some remarks on the wave equation with nonlinear damping and source terms, Nonlinear Analysis T. M. A. 27 (1996), 1165-1175.

R. Ikehata and T. Matsuyama, On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl. 204(1996), 729-753.

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J. 26 (1996), 475-491.

H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Diff. Equations 26(1977), 291-319.

P. Grisvard - Elliptic Problems in Non Smooth Domains, Pitman Publishing Limited, London, 1985.

S. Kesavan - Topics in Functional Analysis and Applications, Willey Easten Limited, New Delhi, 1990.

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Mason-John Wiley, Paris, 1994.

I.Lasiecka and D.Tataru, Uniform Boundary Stabilization of Semilinear Wave Equations with Nonlinear Boundary Damping, Differential and Integral Equations, 6(3), 1993, 507-533.

[LA-TA] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and integral Equations, 6 (1993), 507-533.

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometric conditions, Appl. Math. Optim. 25, (1992), 189-124.

H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porus medium equation backward in time, J. Differential Equations 16, (1974), 319-334.

H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolutions equations with dissipation, Arch. Rational Mech. Anal. 137 (1997), 341-361.

H. A. Levine and A. Smith, A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math. 374, (1987), 1-23.

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIN: Control, Optimisation and Calculus of Variations, 4, (1999), 419-444.

[Ma-So] T. F. Ma and J. A. Soriano, On weak solutions for an evolution equation with exponential nonlinearities, Nonlinear Analysis T. M. A. 37 (1999), 1029-1038.

[Mo] J. Moser, A sharp form of an inequality by Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092.

K. Mochizuki and T. Motai, On energy decay problems for eave equations with nonlinear dissipation term in Rn, J. Math. Soc. Japan 47 (1995), 405-421.

[Mo-Mo] K. Mochizuki and T. Motai, On energy decay problems for wave equations with nonlinear dissipation term in Rn, J. Math. Soc. Japan 47 (1995), 405-421.

[Pi-Ra] D. R. Pitts, M. A. Rammaha, Global existence and non-existence theorems for nonlinear wave equations. Indiana Univ. Math. J. 51 (2002), no. 6, 1479-1509.

[Ra-Str] M. A. Rammaha, T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms. Trans. Amer. Math. Soc. 354 (2002), no. 9, 3621-3637

D. H. Sattiger, On global solutions of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal. 30, 148-172.

J. Serrin, G. Todorova and E. Vitillaro, Existence for a nonlinear wave equation with nonlinear damping and source terms Differential and Integral Equations 16(1), (2003), 13-50.

R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, AMS Providence, 1997.

M. Tsutsumi, Existence and non existence of global solutions for nonlinear parabolic equations, Publ. RIMS, Kyoto Univ. 8 (1972/73), 211-229.

E. Vitillaro, A potential well method for the wave equation with nonlinear source and boundary daming terms, Glasgow Mathematical Journal, 44, (2002), 375-395.

E. Vitillaro, 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.

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186, (2002), 259-298.

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control and Optim. 28, (1990)466-478.

V. Komornik - E. Zuazua, A Direct Method for Boundary Stabilization of the Wave Equation, J. Math. Pures et Appl. 69, (1990), 33-54.

'] J. E. Lagnese, Decay of Solution of Wave Equations in a Bounded Region with Boundary Dissipation, Journal of Differential Equation 50, (1983), 163-182.

'] J. E. Lagnese, Note on Boundary Stabilization of the Wave Equations, Siam J.Control and Optimization, 26(5), (1988), 1250-1257.

J.L. Lions - Problèmes aux Limites dans les Équations aux Dérivées Partielles, Les Presses de l'Université de Montreal, Montreal, 1965.

J.L. Lions and E. Magenes - Problèmes aux Limites non Homogènes, Vol. I, Dunod, Paris, 1968.

J.L. Lions - Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

[Lions] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

[LiMa] J.L.Lions-E.Magenes, Problèmes Aux Limites Non Homogènes et Applications, Dunod, Paris, 1968, Vol. 1.

M. Milla Miranda & L. P. San Gil Jutuca, Existence and boundary stability of solutions for the Kirchhoff equation, Comm. in Partial Diferential Equations, 24(9-10), (1999), 1759-1800.

[Mo] J. Moser, A sharp form of an inequality by Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092.

M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for semilinear dissipative wave equations, Math. Z. 214 (1993), 325-342.

A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., 71 (1992), 455-467.

[Pi-Ra] D. R. Pitts, M. A. Rammaha, Global existence and non-existence theorems for nonlinear wave equations. Indiana Univ. Math. J. 51 (2002), no. 6, 1479-1509.

[Ra-Str] M. A. Rammaha, T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms. Trans. Amer. Math. Soc. 354 (2002), no. 9, 3621-3637

P.A. Raviart and J.M. Thomas - Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles, Masson, Paris, 1983.

'] J. Quinn - D. L. Russell, Asymptotic Stability and Energy Decay for Solutions of Hyperbolic Equations, with boundary damping, Proceedings of the Royal Society of Edinburgh 77(A), (1977), 97-127.

L. Schwartz - Théorie des Distributions, Vol. I, II, Hermann, Paris, 1951.

R. Teman - Navier-Stokes Equations, Theory and Numerical Analysis, NorthHolland, Amsterdan, 1979.

[To1] G. Todorova; E. Vitillaro, Blow-up for nonlinear dissipative wave equations in Rn. J. Math. Anal. Appl. 303 (2005), no. 1, 242-257.

[To2] G. Todorova, Dynamics of non-linear wave equations. Math. Methods Appl. Sci. 27 (2004), no. 15, 1831-1841.

[Tou] D. Toundykov, Optimal decay rates for solutions of nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponents source terms under mixed boundary conditions, Nonlinear Analysis, 2007 (in Press).

[Tru] N. S. Trudinger, On the imbeddings into Orlicz spaces and applications, J. Math. Mech. 17, (1967), 473-484.

[Wi] M. Willem, Minimax Theorems. Birkhäuser, 1996

'] E. Zuazua, Uniform Stabilization of the Wave Equation by Nonlinear Feedback, Siam J. Control and Optimization 28(2), (1990,) 466-477.