Global solvability and blow-up for a nonlinear diffusion model with competing local and memory driven boundary flux

Image restoration is a long-standing problem in low-level computer vision with many interesting applications. We describe a flexible learning framework based on the concept of nonlinear reaction diffusion models for various image restoration problems. By embodying recent improvements in nonlinear diffusion models, we propose a dynamic nonlinear reaction diffusion model with time-dependent parameters (i.e., linear filters and influence functions). In contrast to previous nonlinear diffusion models, all the parameters, including the filters and the influence functions, are simultaneously learned from training data through a loss based approach. We call this approach TNRD-Trainable Nonlinear Reaction Diffusion. The TNRD approach is applicable for a variety of image restoration tasks by incorporating appropriate reaction force. We demonstrate its capabilities with three representative applications, Gaussian image denoising, single image super resolution and JPEG deblocking. Experiments show that our trained nonlinear diffusion models largely benefit from the training of the parameters and finally lead to the best reported performance on common test datasets for the tested applications. Our trained models preserve the structural simplicity of diffusion models and take only a small number of diffusion steps, thus are highly efficient. Moreover, they are also well-suited for parallel computation on GPUs, which makes the inference procedure extremely fast.

Necessary and sufficient conditions for the global solvability of a slow diffusion model with boundary flux governed by memory have been previously shown to be the same as those for a corresponding model with localized nonlinear flux at the boundary. Recent investigations of a similar fast diffusion model with memory have also successfully replicated conditions in parallel with the corresponding localized problem, except for the critical case separating global solvability from blow up in finite time. We provide a suitable modification of an estimate, typically applied to the case of slow diffusion, which also applies to the fast diffusion model and subsequently establishes global solvability in the critical case. Memory terms appearing in the model are of the type which have been introduced in studies of tumor-induced angiogenesis.

A general result on global solvability is established for a diffusion–absorption model with memory-driven flux at the boundary. Such a boundary condition has been studied previously for application to the problem of new capillary growth as induced by a pre-metastatic tumor. In earlier results for the model without absorption, we provided a complete characterization of power law memory boundary conditions regarding either global solvability or blow-up in finite time. It turns out such results are identical to those for the corresponding model with localized power law flux conditions at the boundary. Now in the case of nonlinear absorption added to the model, which in fact better incorporates natural growth factor decay or uptake in the capillary growth application, the threshold of global solvability for the localized model is dependent upon strength of absorption in a way that is not parallel to our result for the memory-driven model. We conclude by proving blow-up results in the radially symmetric case and observing that future studies are needed to complete the full characterization of global solvability.

In this paper, on the basis of the anisotropic diffusion mechanism, analyzes emphatically represented by P - M model of diffusion filter principle of several kinds of nonlinear diffusion model, and their respective characteristics and problems. In-depth analysis of the nonlinear diffusion model, the threshold and termination mechanism of combining image geometric structure feature and visual information (gradient, brightness, contrast, structural information), in view of the existing nonlinear diffusion filtering model, the diffusion coefficient depends on the gradient and the problem that the susceptible to noise interference, presents a fidelity term used in image denoising and restoration contain nonlinear wavelet diffusion model, the theoretical analysis and experimental results show that this method is compared with other diffusion model while denoising can keep image edges and details characteristics, image visual effect is better.

Inthis paper, we study the global uniqueness and solvability of tensor complementarity problems for $\mathcal {H}_{+}$-tensors. We obtain a sufficient condition of the global uniqueness and solvability of tensor complementarity problems for $\mathcal {H}_{+}$-tensors. We present nonlinear dynamical system models for solving the tensor complementarity problem (TCP). We prove that the presented dynamical system models are stable in the sense of Lyapunov stability theory for considering three classes of structured tensors. The computer simulation results further substantiate that the considered dynamical system can be used to solve the TCP.

In this paper we prove the existence, uniqueness and asymptotic behaviour of global regular solutions of the mixed problem for the Kirchho nonlinear model given by the hyperbolic-parabolic equation

We study the characterization of global solvability versus blow up in finite time for a porous medium model including a balance of internal absorption with memory driven flux through the boundary. Such a boundary condition was previously investigated as part of a model for the transmission of tumour-released growth factor from the site of a pre-vascularized tumour up to and across the wall of a nearby capillary, initiating the process of new capillary growth known as angiogenesis. In previous studies of the model without absorption, we have established the characterization of global solvability in a manner that exactly parallels known results for the corresponding model with localized boundary flux conditions. To include models accounting for internal uptake of growth factor, this analysis has recently been extended to a heat equation with absorption, and herein, we consider the case of a porous medium equation with absorption. Conditions for global solvability emerge naturally out of integral estimates and again provide close parallels with results for localized boundary flux models. It is noted that the results provide a complete characterization in a wide range of models considered.

This study explores a three-component chemotaxis model for alopecia areata (AA), incorporating nonlinear diffusion and chemosensitivity functions, as well as general kinetic functions. Under some suitable conditions on the above functions and homogeneous Neumann boundary conditions, we first investigate the global existence and boundedness of classical solutions for the AA system, and find that the higher-order nonlinear diffusion of T cells can prevent the classical solutions of AA system from blow-up when the nonlinear diffusion and chemosensitivity meet the volume filling effect. Additionally, the strong damping effect of T cells can ensure the global boundedness of the solution for AA system irrespective of whether the nonlinear diffusion and chemosensitivity meet the volume filling effect or signal-dependence. In addition, by spectral analysis, we discuss the effect of chemotactic strengths on the stability of AA system around the positive constant equilibrium. By numerical simulations, we analyze the potential causes of AA and propose treatment strategies for future studies.

In this paper, we construct some exact and analytical solutions to a nonlinear diffusion and advection model (Pudasaini in Eng Geol 202: 62–73, 2016) using the Lie symmetry, travelling wave, generalized separation of variables, and boundary layer methods. The model in consideration can be viewed as an extension of viscous Burgers equation, but it describes significantly different physical phenomenon. The nonlinearity in the model is associated with the quadratic diffusion and advection fluxes which are described by the sub-diffusive and sub-advective fluid flow in general porous media and debris material. We also observe that different methods consistently produce similar analytical solutions. This highlights the intrinsic characteristics of the flow of fluid in porous material. The nonlinear diffusion and advection is characterized by a gradually thinning tail that stretches to the rear of the fluid and the evolution of forward advecting frontal bore head, in contrast to the classical linear diffusion and advection. Additionally, we compare solutions for the linear and nonlinear diffusion and advection models highlighting the similarities and differences. The analytical solutions constructed in this paper and the existing high-resolution numerical solution presented previously for the nonlinear diffusion and advection model independently support each other. This implies that the exact and analytical solutions constructed here are physically meaningful and can potentially be applied to calculate the complex nonlinear re-distribution of fluid in porous landscape, and debris and porous materials.

Global Solvability of Nonlinear Diffusion Equations with Forcing at the Boundary

Global solvability and large time behavior to a chemotaxis–haptotaxis model with nonlinear diffusion

We consider again the sixth order Cahn-Hilliard type equation with a nonlinear diffusion, addressed in our previous paper in Commun. Pure Appl. Anal. 10 (2011), 1823--1847. Such PDE arises as a model of oil-water-surfactant mixtures. Applying the approach based on the Backlund transformation and the Leray-Schauder fixed point theorem we generalize the existence result of the above mentioned paper by imposing weaker assumptions on the data. Here we prove the global unique solvability of the problem in the Sobolev space $H^{6,1}(\Omega\times(0,T))$ under the assumption that the initial datum is in $H^3(\Omega)$ whereas previously $H^6(\Omega)$-regularity was required. Moreover, we admit a broarder class of nonlinear terms in the free energy potential.

This paper considers the optimal investment and premium control problem in a diffusion approximation to a non-homogeneous compound Poisson process. In the nonlinear diffusion model, it is assumed that there is an unspecified monotone function describing the relationship between the safety loading of premium and the time-varying claim arrival rate. Hence, in addition to the investment control, the premium rate can be served as a control variable in the optimization problem. Specifically, the problem is investigated in two cases: (i) maximizing the expected utility of terminal wealth, and (ii) minimizing the probability of ruin respectively. In both cases, some properties of the value functions are derived, and closed-form expressions for the optimal policies and the value functions are obtained. The results show that the optimal investment policy and the optimal premium control policy are dependent on each other. Most interestingly, as an example, we show that the nonlinear diffusion model reduces to a diffusion model with a quadratic drift coefficient when the function associated with the premium rate and the claim arrival rate takes a special form. This example shows that the model of study represents a class of nonlinear stochastic control risk model.

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In the present study, we give a very effective nonlinear anisotropic diffusion model to approximate the solution to minimizing the energy functional for additive noise removal. This model is obtained by multiplying the magnitude of the gradient in a nonlinear anisotropic diffusion model. To judge our models, we compared it with the existing model by numerical experiments using explicit numerical schemes with forward-backward diffusivity.