Nonlinear parabolic equations with natural growth in general domains

Dall'aglio, A. ; Giachetti, D. ; Puel, J.-P.

Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 3, pp. 653-683.

Abstract (VO)

We prove an existence result for a class of parabolic problems whose principal part is the $p$-Laplace operator or a more general Leray-Lions type operator, and featuring an additional first order term which grows like $|\nabla u |^{p}$. Here the spatial domain can have infinite measure, and the data may be not regular enough to ensure the boundedness of solutions. As a consequence, solutions are obtained in a class of functions with exponential integrability. An existence result of bounded solutions is also given under additional hypotheses.

Riassunto

In questo articolo si dimostra un risultato di esistenza per una classe di problemi parabolici la cui parte principale è l'operatore $p$-Laplaciano, oppure un operatore più generale del tipo di Leray-Lions, e in cui compare un termine aggiuntivo del primo ordine che cresce come $|\nabla u |^{p}$. Il dominio spaziale in cui si risolve il problema può avere misura infinita, e i dati possono non avere la regolarità necessaria per garantire la limitatezza delle soluzioni. Di conseguenza, si ottengono soluzioni in una classe di funzioni con integrabilità esponenziale. Sotto ipotesi più forti, si prova l'esistenza di soluzioni limitate.

Bibliographie

[1] D. G. Aronson - J. Serrin , Local behavior of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal. 25 (1967), 81-122. | MR | Zbl

[2] D. Blanchard - A. Porretta , Nonlinear parabolic equations with natural growth terms and measure initial data, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 583-622. | fulltext mini-dml | MR | Zbl

[3] L. Boccardo - F. Murat - J.-P. Puel , Existence de solutions non bornées pour certaines équations quasi-linéaires, Portugal. Math., 41, n. 1-4 (1982), 507-534. | fulltext mini-dml | fulltext mini-dml | MR | Zbl

[4] L. Boccardo - F. Murat - J.-P. Puel , Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl., (4) 152 (1988), 183-196. | MR | Zbl

[5] L. Boccardo - F. Murat - J.-P. Puel , Existence results for some quasilinear parabolic equations, Nonlinear Anal., 13 (1989), 373-392. | MR | Zbl

[6] L. Boccardo - F. Murat - J.-P. Puel , $L^1$ estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. (2), 23 (1992), 326-333. | MR | Zbl

[7] L. Boccardo - F. Murat - J.-P. Puel , Résultats d'existence pour certains problémes elliptiques quasilinéaires, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 213-235. | fulltext mini-dml | MR | Zbl

[8] A. Dall'Aglio - V. De Cicco - D. Giachetti - J.-P. Puel , Existence of solutions for nonlinear elliptic equations in unbounded domains, Nonlinear Diff. Eq. and Appl., to appear. | MR | Zbl

[9] A. Dall'Aglio - D. Giachetti - J.-P. Puel , Nonlinear elliptic equations with natural growth in general domains, Ann. Mat. Pura Appl., to appear. | MR | Zbl

[10] A. Dall'Aglio - L. Orsina , Nonlinear parabolic equations with natural growth conditions and $L^1$ data, Nonlin. Anal. TMA, 27, no. 1 (1996), 59-73. | MR | Zbl

[11] E. Di Benedetto , Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. | Zbl

[12] P. Donato - D. Giachetti , Quasilinear elliptic equations with quadratic growth on unbounded domains, Nonlin. Anal., 10 (1986), 791-804. | MR | Zbl

[13] V. Ferone - F. Murat , Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonlinear Anal., 42 (2000), 1309-1326. | MR | Zbl

[14] V. Ferone - M. R. Posteraro - J.-M. Rakotoson , $L^1$-estimates for nonlinear elliptic problems with p-growth in the gradient, J. Ineq. Appl., 3 (1999), 109-125. | MR | Zbl

[15] V. Ferone - M. R. Posteraro - J.-M. Rakotoson , Nonlinear parabolic equations with p-growth and unbounded data, C. R. Acad. Sci. Paris Sér. I Math., 328, no. 4 (1999), 291-296. | MR | Zbl

[16] V. Ferone - M. R. Posteraro - J.-M. Rakotoson , Nonlinear Parabolic Problems with Critical Growth and Unbounded Data, Indiana Math. J. 50, no. 3 (2001), 1201- 1215. | MR | Zbl

[17] N. Grenon , Existence results for some quasilinear parabolic problems, Ann. Mat. Pura Appl., 4 (1993), 281-313. | MR | Zbl

[18] N. Grenon , Asymptotic behaviour for some quasilinear parabolic equations, Nonlinear Anal., 20 (1993), 755-766. | MR | Zbl

[19] R. Landes - V. Mustonen , On parabolic initial-boundary value problems with critical growth for the gradient, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 135-158. | fulltext mini-dml | MR | Zbl

[20] J. Leray - J.L. Lions , Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97- 107. | fulltext mini-dml | MR | Zbl

[21] J.-L. Lions , Quelques mèthodes de rèsolution des problémes aux limites non linèaires, Dunod, Gauthier-Villars, Paris, 1969. | MR | Zbl

[22] F. Nicolosi , Weak solutions of boundary value problems for degenerate parabolic operators in unbounded open sets. Boll. Un. Mat. Ital. 6-C, no. 1 (1985), 269-278. | MR | Zbl

[23] L. Orsina - M.M. Porzio , $L^\infty(Q)$-estimate and existence of solutions for some nonlinear parabolic equations, Boll. U.M.I. 6-B (1992), 631-647. | MR | Zbl

[24] J.-P. Puel , A compactness theorem in quasilinear parabolic problems and application to an existence result, Nonlinear parabolic equations: qualitative properties of solutions (Rome, 1985), Pitman Res. Notes Math. Ser., 149, Longman Sci. Tech., Harlow (1987), 189-199. | MR | Zbl

[25] J. Simon , Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65-96. | MR | Zbl

[26] G. Stampacchia , Equations elliptiques du second ordre à coefficients discontinus. Séminaire de Mathématiques Supérieures, No. 16 Les Presses de l'Université de Montréal, Montrèal, Que. (1966). | MR | Zbl