November  2011, 10(6): 1715-1732. doi: 10.3934/cpaa.2011.10.1715

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

1. 

Universidade Estadual de Campinas, Departamento de Matemática, Campinas, São Paulo, CEP: 13083-859, Brazil

2. 

Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia

Received  July 2010 Revised  February 2011 Published  May 2011

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 $t=0^{+}.$
Citation: Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715
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.   Google Scholar

[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.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

J. Bergh and J. Lofstrom, "Interpolation Spaces,", Springer-Verlag, (1976).   Google Scholar

[4]

L. Boccardo, I. Peral, and M. Escobedo, A Dirichlet problem involving critical exponent,, Journal of Nonlinear Analysis T.M.A., 24 (1995), 1639.  doi: 10.1016/0362-546X(94)E0054-K.  Google Scholar

[5]

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data,, J. Anal. Math., 68 (1996), 277.  doi: 10.1007/BF02790212.  Google Scholar

[6]

M. Cannone and F. Planchon, Self-similar solutions for Navier-Stokes equations in $R^3$,, Comm. Partial Differential Equations, 21 (1996), 179.  doi: 10.1080/03605309608821179.  Google Scholar

[7]

T. Cazenave, F. Dickstein and M. Escobedo, A semilinear heat equation with concave-convex nonlinearity,, Rend. Mat. Appl., 19 (1999), 211.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2010.27.171.  Google Scholar

[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.  doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

[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.  doi: 10.1080/03605309408821042.  Google Scholar

[12]

M. Loayza, The heat equation with singular nonlinearity and singular initial data,, J. Differential Equations, 229 (2006), 509.  doi: 10.1016/j.jde.2006.07.007.  Google Scholar

[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.   Google Scholar

[14]

F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$,, Indiana Univ. Math. J., 29 (1980), 79.  doi: 10.1512/iumj.1980.29.29007.  Google Scholar

[15]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29.  doi: 10.1007/BF02761845.  Google Scholar

[16]

M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ spaces with time-dependent external force,, Math. Ann., 317 (2000), 635.  doi: 10.1007/PL00004418.  Google Scholar

show all references

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.   Google Scholar

[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.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

J. Bergh and J. Lofstrom, "Interpolation Spaces,", Springer-Verlag, (1976).   Google Scholar

[4]

L. Boccardo, I. Peral, and M. Escobedo, A Dirichlet problem involving critical exponent,, Journal of Nonlinear Analysis T.M.A., 24 (1995), 1639.  doi: 10.1016/0362-546X(94)E0054-K.  Google Scholar

[5]

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data,, J. Anal. Math., 68 (1996), 277.  doi: 10.1007/BF02790212.  Google Scholar

[6]

M. Cannone and F. Planchon, Self-similar solutions for Navier-Stokes equations in $R^3$,, Comm. Partial Differential Equations, 21 (1996), 179.  doi: 10.1080/03605309608821179.  Google Scholar

[7]

T. Cazenave, F. Dickstein and M. Escobedo, A semilinear heat equation with concave-convex nonlinearity,, Rend. Mat. Appl., 19 (1999), 211.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2010.27.171.  Google Scholar

[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.  doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

[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.  doi: 10.1080/03605309408821042.  Google Scholar

[12]

M. Loayza, The heat equation with singular nonlinearity and singular initial data,, J. Differential Equations, 229 (2006), 509.  doi: 10.1016/j.jde.2006.07.007.  Google Scholar

[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.   Google Scholar

[14]

F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$,, Indiana Univ. Math. J., 29 (1980), 79.  doi: 10.1512/iumj.1980.29.29007.  Google Scholar

[15]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29.  doi: 10.1007/BF02761845.  Google Scholar

[16]

M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ spaces with time-dependent external force,, Math. Ann., 317 (2000), 635.  doi: 10.1007/PL00004418.  Google Scholar

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