Lectures on Nonlinear Evolution Equations

Global solution to the Cauchy problem of nonlinear thermodiffusion in a solid body

“…We apply Friedrichs mollifiers (cf. [15]) in the following form. We put U ε = J ε Ū , θ ε = J ε θ , where J ε (x) is a nonnegative function that belongs to C ∞ 0 and is convolution.…”

“…The proof is based on Theorem 2.4 (cf. [7], [15]). Below, we show that the assumptions of that theorem are satisfied:…”

Inverse scattering for the non‐linear Schrödinger equation: Reconstruction of the potential and the non‐linearity

“…This is the reason why we only have to assume that (1.2) holds for |u| ≤ γ. For results on scattering for the nonlinear Schrödinger equation in the case where V 0 = 0 see [16], [17], [18], [10], [9], [11], [3], [7], [2] and the references quoted there. In [8] the direct scattering for (1.1) with F = F (u) was studied for n ≥ 3.…”

𝐿^{𝑝}-𝐿^{𝑞} decay estimates for the Cauchy problem of linear thermoelastic systems with second sound in one space variable

“…They are hyperbolic-parabolic coupled systems ( [1]- [3], [6], [9], [19]). Although the classical thermoelasticity theory, based on the Fourier law of heat conduction, predicts that a thermal disturbance of material conducting heat will be propagated with infinite speeds, experiments on certain dielectric crystals at very low temperature have produced results that show the thermal disturbance is transmitted as "wave-like" pulses with finite speeds (see [2], [3], and references therein).…”