On the heat equation with concave-convex nonlinearity and initial data inweak-$L^p$ spaces

Lucas C. F. Ferreira; Elder J. Villamizar-Roa

doi:10.3934/cpaa.2011.10.1715

Article Contents

2011, Volume 10, Issue 6: 1715-1732. Doi: 10.3934/cpaa.2011.10.1715

This issue Previous Article Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations Next Article Blow-up rates of large solutions for semilinear elliptic equations

On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces

Lucas C. F. Ferreira^1, and
Elder J. Villamizar-Roa^2,

1.
Universidade Estadual de Campinas, Departamento de Matemática, Campinas, São Paulo, CEP: 13083-859
2.
Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga

Received: July 2010

Revised: February 2011

Published: November 2011

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Abstract

We analyze the local well-posedness of the initial-boundary value problem for the heat equation with nonlinearity presenting a combined concave-convex structure and taking the initial data in weak-$L^{p}$ spaces. Moreover we give a new uniqueness class and obtain some results about the behavior of the solutions near $t=0^{+}.$

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Mathematics Subject Classification: Primary: 35K05, 35K55, 35A05; Secondary: 42B35.

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References

[1]	J. Aguirre and M. Escobedo, A Cauchy problem for $u_t-\Delta u=u^p$ with $0, Ann. Fac. Sci. Toulouse Math., 8 (1986), 175-203.
[2]	A. Ambrosetti, H. Brezis, and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.doi: 10.1006/jfan.1994.1078.
[3]	J. Bergh and J. Lofstrom, "Interpolation Spaces," Springer-Verlag, New York, 1976.
[4]	L. Boccardo, I. Peral, and M. Escobedo, A Dirichlet problem involving critical exponent, Journal of Nonlinear Analysis T.M.A., 24 (1995), 1639-1648.doi: 10.1016/0362-546X(94)E0054-K.
[5]	H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304.doi: 10.1007/BF02790212.
[6]	M. Cannone and F. Planchon, Self-similar solutions for Navier-Stokes equations in $R^3$, Comm. Partial Differential Equations, 21 (1996), 179-193.doi: 10.1080/03605309608821179.
[7]	T. Cazenave, F. Dickstein and M. Escobedo, A semilinear heat equation with concave-convex nonlinearity, Rend. Mat. Appl., 19 (1999), 211-242.
[8]	L. C. F. Ferreira and E. J. Villamizar-Roa, Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential and Integral Equations, 19 (2006), 1349-1370.
[9]	L. C. F. Ferreira and E. J. Villamizar-Roa, On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^p$ spaces, Discrete Contin. Dyn. Syst., 27 (2010), 171-183.doi: 10.3934/dcds.2010.27.171.
[10]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system., J. Differential Equations, 62 (1986), 186-212.doi: 10.1016/0022-0396(86)90096-3.
[11]	H. Kozono and Y. Yamazaki, Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data, Comm. P.D.E., 19 (1994), 959-1014.doi: 10.1080/03605309408821042.
[12]	M. Loayza, The heat equation with singular nonlinearity and singular initial data, J. Differential Equations, 229 (2006), 509-528.doi: 10.1016/j.jde.2006.07.007.
[13]	Y. Maekawa and T. Terasawa, The Navier-Stokes equations with initial data in uniformly local $L^p$ spaces, Differential Integral Equations, 19 (2006), 369-400.
[14]	F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102.doi: 10.1512/iumj.1980.29.29007.
[15]	F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.doi: 10.1007/BF02761845.
[16]	M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ spaces with time-dependent external force, Math. Ann., 317 (2000), 635-675.doi: 10.1007/PL00004418.