A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations

Abstract

A degenerate parabolic partial differential equation with a time derivative and first-and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

Multiple positive solutions to a singular boundary value problem for a superlinear Emden–Fowler equation

“…Consequently, for this range of values of ζ, the solutions w ζ gives a family of monotonic increasing-decreasing solutions to (12), (13) for Γ > Γ . In summary, the main result is that problem (1), (2) has multiple monotonic increasing and increasing-decreasing solutions such that dy dx (1) takes place in the interval…”

“…In the theory of the integral equation [13][14][15] it is known that either problem (17), (18) has no solution, or it has one parameter family of solutions which contains a (unique) solution called "maximal" solution satisfying (21). We recall briefly some relevant results of [13][14][15] concerning problem (17), (18).…”

“…We recall briefly some relevant results of [13][14][15] concerning problem (17), (18). It is shown that there exists a minimal Γ = Γ (σ ) > 0 such that (17) (20).…”

“…In the remainder of this section we shall obtain some estimates for ϕ(1) and Σ , or Γ . It turns out that the real number ϕ(1) = −w (0) plays the role of the shooting parameter for problem (12), (13). So, one advantage gained from the integral equation (17) (or the Abel equation) is that the estimate of ϕ(1) will be easily deduced.…”

“…To obtain an estimate from above for ϕ (1) or to obtain increasing-decreasing solutions, we look for a solution to (12), (13) such that ζ = w (0) > 0, ζ being the shooting parameter. In Section 2 it is argued that for any ζ = 0 the (unique) local solution to (12) with the initial condition w(0) = 1, w (0) = ζ, is global and tends to 0 at infinity.…”