On very singular similarity solutions of a higher-order semilinear parabolic equationResearch suppor

We study the large-time behaviour of solutions of a semilinear 2mth-order parabolic equation 1,\end{equation*} \] SRC=http://ej.iop.org/images/0951-7715/17/3/017/non172874ude001.gif/> with bounded integrable initial data u0 decaying exponentially at infinity. For the semilinear heat equation (m = 1), the asymptotic behaviour was established in detail in the 1980s. Our main goal is to justify that, for any m ≥ 1 in the subcritical range 1 < p < p0 = 1 + (2m/N), there exists a finite number, M ~ N(p0 − p)/2(p − 1) → ∞ as p → 1+, of different very singular self-similar solutions of the form where each V is a radial, exponentially decaying solution of the elliptic equation By a perturbation technique, we establish the existence of radially symmetric very singular solution profiles Vl for p close to critical bifurcation exponents pl = 1 + (2m/(l + N)), l = 0, 2, ..., where the first one, V0, is shown to be stable. Discrete and countable subsets of other self-similar and approximately self-similar patterns are introduced.

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

[2]  J. M. Ball,et al.  GEOMETRIC THEORY OF SEMILINEAR PARABOLIC EQUATIONS (Lecture Notes in Mathematics, 840) , 1982 .

[3]  Antti Kupiainen,et al.  Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations , 1993, chao-dyn/9306008.

[4]  J. Bricmont,et al.  Stable nongaussian diffusive profiles , 1996 .

[5]  L. Peletier,et al.  Spatial Patterns: Higher Order Models in Physics and Mechanics , 2001 .

[6]  L. Véron,et al.  Existence and uniqueness of the very singular solution of the porous media equation with absorption , 1988 .

[7]  Victor A. Galaktionov,et al.  Critical global asymptotics in higher-order semilinear parabolicequations , 2003 .

[8]  U. Elias.  Eigenvalue problems for the equation Ly + λp(x) y = 0 , 1978 .

[9]  B. Rynne.  Global bifurcation for 2mth-order boundary value problems and infinitely many solutions of superlinear problems , 2003 .

[10]  J. F. Williams,et al.  Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations , 2004, SIAM J. Appl. Math..

[11]  K. Deimling.  Nonlinear functional analysis , 1985 .

[12]  M. Fedoryuk.  SINGULARITIES OF THE KERNELS OF FOURIER INTEGRAL OPERATORS AND THE ASYMPTOTIC BEHAVIOUR OF THE SOLUTION OF THE MIXED PROBLEM , 1977 .

[13]  J. F. Williams,et al.  Blow-up and global asymptotics of the unstable Cahn-Hilliard equation with a homogeneous nonlinearity , 2006 .

[14]  Daniel B. Henry,et al.  Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations , 1985 .

[15]  M. Kreĭn,et al.  Stability of Solutions of Differential Equations in Banach Spaces , 1974 .

[16]  M. Solomjak,et al.  Spectral Theory of Self-Adjoint Operators in Hilbert Space , 1987 .

[17]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[18]  A. Friedman,et al.  Nonlinear Parabolic Equations Involving Measures as Initial Conditions , 1981 .

[19]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[20]  M. A. Krasnoselʹskii.  Topological methods in the theory of nonlinear integral equations , 1968 .

[21]  L. Peletier,et al.  A very singular solution of the heat equation with absorption , 1986 .

[22]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[23]  L. Peletier,et al.  Singular solutions of the heat equation with absorption , 1985 .

[24]  Victor A. Galaktionov,et al.  Global solutions of higher-order semilinear parabolic equations in the supercritical range , 2002, Advances in Differential Equations.

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

[26]  Victor A. Galaktionov,et al.  On a spectrum of blow–up patterns for a higher–order semilinear parabolic equation , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[27]  Sigurd B. Angenent,et al.  The Morse-Smale property for a semi-linear parabolic equation , 1986 .

[28]  V. Galaktionov,et al.  ON ASYMPTOTIC “EIGENFUNCTIONS” OF THE CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION , 1986 .

[29]  Michael E. Taylor,et al.  Partial Differential Equations III , 1996 .

[30]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[31]  M. A. Krasnoselʹskii,et al.  Geometrical Methods of Nonlinear Analysis , 1984 .

[32]  Avner Friedman,et al.  Partial differential equations , 1969 .

[33]  L. A. Peletier,et al.  Large time behaviour of solutions of the porous media equation with absorption , 1986 .

[34]  Victor A. Galaktionov,et al.  A Stability Technique for Evolution Partial Differential Equations , 2009 .

[35]  Charles M. Elliott,et al.  On the Cahn-Hilliard equation , 1986 .

[36]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[37]  Mary C. Pugh,et al.  Long-wave instabilities and saturation in thin film equations , 1998 .

[38]  A. P. Mikhailov,et al.  Blow-Up in Quasilinear Parabolic Equations , 1995 .

[39]  Junping Shi,et al.  Morse indices and exact multiplicity of solutions to semilinear elliptic problems , 1999 .

[40]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[41]  L. Segel,et al.  Nonlinear aspects of the Cahn-Hilliard equation , 1984 .