On the Diffusion Phenomenon of Quasilinear Hyperbolic Waves

L1 Decay estimates for dissipative wave equations

“…Among the applications of this result, we mention an improvement of some of the estimates which can be used to determine the critical exponent of a kind of dissipative wave equation with non-linear dissipation and source term, as well as the asymptotic behaviour of its solutions when these exist globally (see Todorova and Yordanov [1], for details). Related to this, we can also considerably simplify the proof of the di usion phenomenon of hyperbolic waves we gave in Reference [2].…”

“…in L p (R n ); 16p6+∞, (note that the dissipation in (5) is stronger than in (3)), and discusses their similarity to the decay rates of the purely di usive parabolic problem; in Reference [4], Racke establishes decay rates in L p (R n ), 26p6+∞, for solutions of problem (3) [8]). As a ÿnal example, in our above-mentioned paper [2] we have described the di usion phenomenon for (3), whereby the di erence of the solutions to (3) and (4) decays faster than either one, and have extended this result to a quasilinear version of (3) and (4).…”

Critical regularity of nonlinearities in semilinear classical damped wave equations

“…The diffusion phenomenon between linear heat and linear classical damped wave models (see [3], [6], [8] and [9]) explains the parabolic character of classical damped wave models with power nonlinearities from the point of decay estimates of solutions. In the mathematical literature (see for instance [1]) the situation is in general described as follows: We have a semilinear Cauchy problem…”

“…Our goal is to discuss the influence of the function µ on the global (in time) existence of small data Sobolev solutions or on statements for blow-up of Sobolev solutions to (3). In the following result, we assume that the modulus of continuity µ given in (3) satisfies the following two conditions:…”

The asymptotic behavior of the Bresse–Cattaneo system

“…We should note here that the dissipative effects introduced by the Cattaneo law are usually weaker than those induced by the Fourier law, but when time evolves, the solutions of both systems (1.3) and (1.4) behave similarly and with the same decay rate. See [6,13,18].…”