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  • 1
    Electronic Resource
    Electronic Resource
    Maximal sign-invariants of quasilinear parabolic equations with gradient diffusivity (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 4948-4964 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We consider quasilinear parabolic equations with gradient diffusivity ut=div(|∇u|σ∇u)+f(u), x∈RN, t〉0, where σ≠−1 is a fixed constant and f(u) is a given smooth function. We also study quasilinear parabolic equations with a gradient-dependent coefficient ut=h(|∇u|)Δu+f(u), with a smooth function h(p). For both classes of equations we derive first-order sign-invariants, i.e., first-order operators preserving their signs on the evolution orbits {u(⋅,t),t〉0}. We give a complete description of (maximal) sign-invariants of prescribed structures. As a consequence, we construct new exact solutions of some quasilinear equations. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532501
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  • 2
    Electronic Resource
    Electronic Resource
    On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation (1996)
    Springer
    Nonlinear differential equations and applications 3 (1996), S. 269-286 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ℝ N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u p with a fixedp〉1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v p defined for allv≫1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01194067
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  • 3
    Electronic Resource
    Electronic Resource
    Geometrical properties of the solutions of one-dimensional nonlinear parabolic equations (1995)
    Springer
    Mathematische Annalen 303 (1995), S. 741-769 
    Details
    ISSN: 1432-1807
    Keywords: 35K55 ; 35K65
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01461014
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  • 4
    Electronic Resource
    Electronic Resource
    Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations (1995)
    Springer
    Archive for rational mechanics and analysis 129 (1995), S. 225-244 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We characterize the occurrence of complete or incomplete blow-up (and extinction) for a general quasilinear heat equation of the form $$(HE){\text{ }}u_t = (\phi (u))_{xx} \pm f(u){\text{ in }}R \times (0,T)$$ in terms of the constitutive functions φ and f We assume that φ′(u)〉0 for u〉0 and that f(u)≧0. For the positive sign + before f(u) in (HE), with f(u) superlinear as u→∞, blow-up occurs in finite time: sup x u(x, t)→∞ as t→T〈∞. For the negative sign, we consider the case of singular absorption: f(u)→∞ as u→0. Then initially positive solutions vanish at some point in finite time (extinction), and a singularity in the equation occurs there. An important aspect of blow-up or extinction problems is the possibility of having a nontrivial extension of the solution for t〉T, i.e., after the singularity occurs. If such continuation exists, we say that the blow-up (extinction) is incomplete; otherwise it is called complete. Our characterization is based on the qualitative behaviour of the family of travelling-wave solutions and a proper use of the Intersection-Comparison argument. The analysis applies to other nonlinear models, like the equations with gradient-dependent diffusivity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00383674
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  • 5
    Electronic Resource
    Electronic Resource
    Regularity of Interfaces in Diffusion Processes under the Influence of Strong Absorption (1999)
    Springer
    Archive for rational mechanics and analysis 149 (1999), S. 183-212 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract .We present a method of analysis which allows us to establish the interface equation and to prove Lipschitz continuity of interfaces and solutions which appear in a large class of nonlinear parabolic equations and conservation laws posed in one space dimension. Its main feature is intersection comparison with travelling waves. The method is explained on the following study case: We consider the Cauchy problem for the diffusion-absorption model: $$ u_t = \left( u^m \right)_{xx} - u^p, \quad u\ge 0, $$ in the range of parameters $m〉1,\ 0〈p〈1,\ m+p\ge2$, i.e., we have slow diffusion combined with strong absorption. Contrary to the case $p\ge 1$ , or the purely diffusive equation $u_t=(u^m)_{xx},\ m〉1$ , where the support of the solution expands with time and the motion is governed by Darcy's law, in the strong absorption range there might appear shrinking interfaces and the interface evolution obeys a different mechanism. Previous methods have failed to provide an adequate analysis of the interface motion and regularity in such a situation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002050050172
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  • 6
    Electronic Resource
    Electronic Resource
    Asymptotic Behaviour near Finite-Time Extinction for the Fast Diffusion Equation (1997)
    Springer
    Archive for rational mechanics and analysis 139 (1997), S. 83-98 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We study the Cauchy problem for the fast diffusion equation $$u_t = \Delta (u^m), u \gee 0 \qquad \hbox { in } {\bf R} ^N \times {\bf R} _+, $$ when $N \gee 3$ and $0〈 m 〈 {N-2 \over N} $ . For a class of nonnegative radially symmetric finite-mass solutions, which vanish identically at a given time $T〉 0$ , we show that their asymptotic behaviour as $ t \nearrow T$ is described by a uniquely determined self-similar solution of the second kind: $u_*(r,t) = (T-t)^\alpha f(\eta )$ , where $\eta =r/(T-t)^\beta $ and $\alpha ={1-2\beta \over 1-m} $ and $r=|x|$ . Here $\beta $ is determined from a nonlinear eigenvalue problem involving an ordinary differential equation for the function $f$ . Special attention is paid to the case when $m ={N-2 \over N+2} $ . Then $\beta =0$ and the function $f$ can be found explicity. The proof is based on a geometric Lyapunov-type argument and comparison arguments based on the intersection properties of the solution graphs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002050050048
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