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Asymptotic behavior of solutions for linear parabolic equations with general measure data. (English) Zbl 1124.35318
Summary: In this note we deal with the asymptotic behavior as $t$ tends to infinity of solutions for linear parabolic equations whose model is $$\cases u_t-\Delta u=\mu &\text{in }(0,T)\times\Omega,\\ u(0,x)= u_0 &\text{in }\Omega,\endcases$$ where $\mu$ is a general, possibly singular, Radon measure which does not depend on time, and $u_0\in L^1(\Omega)$. We prove that the duality solution, which exists and is unique, converges to the duality solution (as introduced by {\it G. Stampacchia} [Ann. Inst. Fourier 15, No. 1, 189--257 (1965) and Colloques Int. Centre nat. Rech. Sci. 146, 189--258 (1965; Zbl 0151.15401)]) of the associated elliptic problem.

35K15Second order parabolic equations, initial value problems
35R05PDEs with discontinuous coefficients or data
35B40Asymptotic behavior of solutions of PDE
Radon measure
Full Text: DOI arXiv
[1] Arosio, A.: Asymptotic behavior as t$\to +\infty $ of solutions of linear parabolic equations with discontinuous coefficients in a bounded domain. Comm. partial differential equations 4, No. 7, 769-794 (1979) · Zbl 0436.35014
[2] Boccardo, L.; Dall’aglio, A.; Gallouët, T.; Orsina, L.: Nonlinear parabolic equations with measure data. J. funct. Anal. 147, No. 1, 237-258 (1997) · Zbl 0887.35082
[3] Dal Maso, G.; Murat, F.; Orsina, L.; Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. scuola norm. Sup. Pisa cl. Sci. 28, 741-808 (1999) · Zbl 0958.35045
[4] Friedman, A.: Partial differential equations of parabolic type. (1964) · Zbl 0144.34903
[5] Ladyzhenskaja, O. A.; Solonnikov, V.; Uraltceva, N. N.: Linear and quasilinear parabolic equations. (1970)
[6] Leonori, T.; Petitta, F.: Asymptotic behavior of solutions for parabolic equations with natural growth term and irregular data. Asymptotic anal. 48, No. 3, 219-233 (2006) · Zbl 1145.35346
[7] F. Petitta, Asymptotic behavior of solutions for parabolic operators of Leray -- Lions type and measure data, in press · Zbl 1152.35323
[8] F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data, in press · Zbl 1150.35060
[9] Porretta, A.: Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. mat. Pura appl. (IV) 177, 143-172 (1999) · Zbl 0957.35066
[10] Prignet, A.: Existence and uniqueness of entropy solutions of parabolic problems with L1 data. Nonlinear anal. TMA 28, 1943-1954 (1997) · Zbl 0909.35075
[11] Spagnolo, S.: Convergence de solutions d’équations d’évolution. Proceedings of the international meeting on recent methods in nonlinear analysis, 311-327 (1979)
[12] Stampacchia, G.: Le problème de Dirichlet pour LES équations elliptiques du seconde ordre à coefficientes discontinus. Ann. inst. Fourier (Grenoble) 15, 189-258 (1965) · Zbl 0151.15401
[13] Trudinger, N. S.: On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. pure appl. Math. 20, 721-747 (1967) · Zbl 0153.42703