# American Institute of Mathematical Sciences

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
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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. Google Scholar [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. 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), 369-400. Google Scholar [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. Google Scholar [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. Google Scholar [16] M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$spaces with time-dependent external force, Math. Ann., 317 (2000), 635-675. 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), 175-203.  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), 519-543. doi: 10.1006/jfan.1994.1078.  Google Scholar [3] J. Bergh and J. Lofstrom, "Interpolation Spaces," Springer-Verlag, New York, 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), 1639-1648. 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), 277-304. 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), 179-193. 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), 211-242.  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), 1349-1370.  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), 171-183. 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), 186-212. 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), 959-1014. doi: 10.1080/03605309408821042.  Google Scholar [12] M. Loayza, The heat equation with singular nonlinearity and singular initial data, J. Differential Equations, 229 (2006), 509-528. 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), 369-400.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ spaces with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.  Google Scholar
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