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LYAPUNOV FUNCTIONALS IN SINGULAR LIMITS FOR PERTURBED QUASILINEAR DEGENERATE PARABOLIC EQUATIONS
Preview AbstractAs a key example, we study the asymptotic behaviour near finite focusing time t=T of radial solutions of the porous medium equation with absorption
with bounded compactly supported initial data u(x,0)=u0(|x|), and exponents m>1 and p>pc, where pc=pc(m,N)∈(-m,0) is a critical exponent. We show that under certain assumptions, the behaviour of the solution as t→T- near the origin is described by self-similar Graveleau solutions of the porous medium equation ut=Δum. In the rescaled variables, we deal with an exponential non-autonomous perturbation of a quasilinear parabolic equation, which is shown to admit an approximate Lyapunov functional. The result is optimal, and in the critical case p=pc an extra ln(T-t) scaling of the Graveleau asymptotics is shown to occur. Other types of self-similar and non self-similar focusing patterns are discussed.
BLOW-UP SOLUTIONS OF THE TWO-DIMENSIONAL HEAT EQUATION DUE TO A LOCALIZED MOVING SOURCE
Preview AbstractThe problem examined is that of a localized energy source which undergoes planar motion along the surface of a reactive-diffusive medium. This is representative of a laser beam that is moving across the flat surface of a combustible material during a cutting, welding or heat treating process. The mathematical model for this situation is a heat equation in two-dimensions with a nonlinear source term, which is localized around a reference point that is allowed to move. Results are derived that indicate the roles played by the size, strength and motion of the localized source in determining whether or not a blow-up occurs.
Blowup for a Kirchhoff-type parabolic equation with logarithmic nonlinearity
Preview AbstractIn this paper, we consider a Kirchhoff-type parabolic equation with logarithmic nonlinearity. By making a more general assumption about the Kirchhoff function, we establish a new finite time blow-up criterion. In particular, the blow-up rate and the upper and lower bounds of the blow-up time are also derived. These results generalize some recent ones in which the blow-up results were obtained when the Kirchhoff function was assumed to be a very special form.