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Remarks on a nonlocal problem involving the Dirichlet energy. (English) Zbl 1117.35034
The authors study the following nonlinear parabolic problem of nonlocal type depending on the Dirichlet integral: find \(u=u\left( x,t\right) \) solution to \[ u_t-a\left( \int_\Omega \left| \nabla u\right| ^2\,dx\right) \Delta u=f\;\;in\;\Omega \times \mathbb R^+, \]
\[ u\left( x,0\right) =u_0\left( x\right) \;in\;\Omega ,\;\;u=0\;on\;\partial \Omega \times R^{+}, \] where \(\Omega \) is a bounded, smooth open subset of \(\mathbb R^n\), \(n\geq 1\), \(f\in L^2( \Omega )\), \(u_0\in H_0^1(\Omega)\) and \(a=a(s)\) is a continuous function such that \(0<m\leq a\left( s\right) \leq M\). After proving the uniqueness and existence of the solution, the authors study the corresponding stationary problem and finally give some results concerning the asymptotic behaviour of the solution.

35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35J20 Variational methods for second-order elliptic equations
35R10 Partial functional-differential equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: EuDML
[1] T. CAZENAVE - A. HARAUX, An Introduction to Semilinear Evolution Equations, Oxford Science Publications (1998). Zbl0926.35049 MR1691574 · Zbl 0926.35049
[2] N.-H. CHANG - M. CHIPOT, On some mixed boundary value problems with nonlocal diffusion. (To appear in Adv. in Math. Sci. and Appl.). Zbl1064.35083 MR2083613 · Zbl 1064.35083
[3] N.-H. CHANG - M. CHIPOT, On some model diffusion problems with a nonlocal lower order term. (To appear in the Chinese Ann. of Math.). Zbl1039.35056 MR1982061 · Zbl 1039.35056
[4] N.-H. CHANG - M. CHIPOT, Nonlinear nonlocal evolution problems. (To appear in RACSAM). Zbl1067.35035 MR2126241 · Zbl 1067.35035
[5] M. CHIPOT, Elements of Nonlinear Analysis, Birkhäuser Advanced Text (2000). Zbl0964.35002 MR1801735 · Zbl 0964.35002
[6] M. CHIPOT - B. LOVAT, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, DCDIS, Series A, 8 (2001), J 1, pp. 35-51. Zbl0984.35066 MR1820664 · Zbl 0984.35066
[7] M. CHIPOT - B. LOVAT, On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), pp. 65-81. Zbl0921.35071 MR1675465 · Zbl 0921.35071
[8] M. CHIPOT, L. MOLINET: Asymptotic behavior of some nonlocal diffusion problems. Applicable Analysis, 80 (2001) p. 273-315. Zbl1023.35016 MR1914683 · Zbl 1023.35016
[9] M. CHIPOT - J. F. RODRIGUES, On a class of nonlocal nonlinear elliptic problems, Math. Mod. and Num. Anal., 26 (1992), pp. 447-468. Zbl0765.35021 MR1160135 · Zbl 0765.35021
[10] M. CHIPOT - M. SIEGWART, On the asymptotic behaviour of some nonlocal mixed boundary value problems. To appear. Zbl1052.35100 MR2060227 · Zbl 1052.35100
[11] R. DAUTRAY - J. L. LIONS, Mathematical analysis and numerical methods for science and technology, Springer Verlag (1990). · Zbl 0784.73001
[12] D. GILBARG - N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer Verlag, New York (1983). Zbl0562.35001 MR737190 · Zbl 0562.35001
[13] A. HARAUX, Systèmes Dynamiques Dissipatifs et Applications, Masson Paris (1991). Zbl0726.58001 MR1084372 · Zbl 0726.58001
[14] D. KINDERLEHRER - G. STAMPACCHIA, An introduction to variational inequalities and their applications, Acad. Press (1980). Zbl0457.35001 MR567696 · Zbl 0457.35001
[15] J. L. LIONS, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthiers-Villards, Paris (1969). Zbl0189.40603 MR259693 · Zbl 0189.40603
[16] M. SIEGWART, Thesis, University of Zürich (to appear).
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