Global well-posedness for fractional Sobolev-Galpern type equations

<p style='text-indent:20px;'>This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.</p>

Gradient nonlinearity correction in liver DWI using motion-compensated diffusion encoding waveforms

Objective To experimentally characterize the effectiveness of a gradient nonlinearity correction method in removing ADC bias for different motion-compensated diffusion encoding waveforms. Methods The diffusion encoding waveforms used were the standard monopolar Stejskal–Tanner pulsed gradient spin echo (pgse) waveform, the symmetric bipolar velocity-compensated waveform (sym-vc), the asymmetric bipolar velocity-compensated waveform (asym-vc) and the asymmetric bipolar partial velocity-compensated waveform (asym-pvc). The effectiveness of the gradient nonlinearity correction method using the spherical harmonic expansion of the gradient coil field was tested with the aforementioned waveforms in a phantom and in four healthy subjects. Results The gradient nonlinearity correction method reduced the ADC bias in the phantom experiments for all used waveforms. The range of the ADC values over a distance of ± 67.2 mm from isocenter reduced from 1.29 × 10–4 to 0.32 × 10–4 mm2/s for pgse, 1.04 × 10–4 to 0.22 × 10–4 mm2/s for sym-vc, 1.22 × 10–4 to 0.24 × 10–4 mm2/s for asym-vc and 1.07 × 10–4 to 0.11 × 10–4 mm2/s for asym-pvc. The in vivo results showed that ADC overestimation due to motion or bright vessels can be increased even further by the gradient nonlinearity correction. Conclusion The investigated gradient nonlinearity correction method can be used effectively with various motion-compensated diffusion encoding waveforms. In coronal liver DWI, ADC errors caused by motion and residual vessel signal can be increased even further by the gradient nonlinearity correction.

On the extinction problem for a p-Laplacian equation with a nonlinear gradient source

We deal with the extinction properties of the weak solutions for a p-Laplacian equation with a gradient nonlinearity. The critical extinction exponent is specified and the decay estimates of the extinction solutions are given.

On the Jordan–Moore–Gibson–Thompson Wave Equation in Hereditary Fluids with Quadratic Gradient Nonlinearity

We prove global solvability of the third-order in time Jordan–More–Gibson–Thompson acoustic wave equation with memory in $${\mathbb {R}}^n$$ R n , where $$n \ge 3$$ n ≥ 3 . This wave equation models ultrasonic propagation in relaxing hereditary fluids and incorporates both local and cumulative nonlinear effects. The proof of global existence is based on a sequence of high-order energy bounds that are uniform in time, and derived under the assumption of an exponentially decaying memory kernel and sufficiently small and regular initial data.