Positive solution to semilinear parabolic equation associated with critical Sobolev exponent

We study the semilinear parabolic equation $${u_{t}- \Delta u = u^{p}, u \geq 0}$$ on the whole space RN, $${N \geq 3}$$ associated with the critical Sobolev exponent p = (N + 2)/(N − 2). Similarly to the bounded domain case, there is threshold blowup modulus concerning the blowup in finite time. Furthermore, global in time behavior of the threshold solution is prescribed in connection with the energy level, blowup rate, and symmetry.

[1]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[2]  Laurent Veron,et al.  Singularities of solutions of second order quasilinear equations , 1996 .

[3]  Pierre-Louis Lions,et al.  The concentration-compactness principle in the Calculus of Variations , 1985 .

[4]  Peter Polácik,et al.  Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation , 2008, Asymptot. Anal..

[5]  Otared Kavian,et al.  Remarks on the large time behaviour of a nonlinear diffusion equation , 1987 .

[6]  P. Quittner.  The decay of global solutions of a semilinear heat equation , 2008 .

[7]  Tadashi Kawanago,et al.  Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity , 1996 .

[8]  A. Friedman,et al.  Blow-up of positive solutions of semilinear heat equations , 1985 .

[9]  Wenxiong Chen,et al.  Classification of solutions of some nonlinear elliptic equations , 1991 .

[10]  Takashi Suzuki,et al.  Semilinear parabolic equation in RN associated with critical Sobolev exponent , 2010 .

[11]  P. Lions,et al.  Solutions globales d'équations de la chaleur semi linéaires , 1984 .

[12]  Michael Struwe,et al.  A global compactness result for elliptic boundary value problems involving limiting nonlinearities , 1984 .

[13]  Otared Kavian,et al.  Variational problems related to self-similar solutions of the heat equation , 1987 .

[14]  P. Lions.  Asymptotic behavior of some nonlinear heat equations , 1982 .

[15]  Takashi Suzuki.  Semilinear parabolic equation on bounded domain with critical Sobolev exponent , 2008 .

[16]  Stathis Filippas,et al.  Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  Jean-Michel Coron,et al.  Convergence of solutions of H-systems or how to blow bubbles , 1985 .

[18]  Basilis Gidas,et al.  Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth , 1989 .

[19]  M. Ishiwata.  On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent , 2010 .

[20]  Victor A. Galaktionov,et al.  Continuation of blowup solutions of nonlinear heat equations in several space dimensions , 1997 .