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We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time

We consider the semilinear heat equation ∂tu-Δu=f(u),(x,t)∈RN×[0,T),(1)with f(u)=|u|p-1uloga(2+u2), where p>1 is Sobolev subcritical and a∈R. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely u′=|u|p-1uloga(2+u2). In other words, all blow-up solutions in the Sobolev subcritical range are Type I solutions. To the best of our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.

We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution u(x,t), the graph x → T(x) of its blow-up points and S ⊂ ℝ the set of all characteristic points and show that S has an empty interior. Finally, given x 0 ∈ S, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that T(x) forms a corner of angle $\\frac{\\mathrm{\\pi }}{2}$ at x 0 .

Crohn’s disease is an inflammatory bowel disease (IBD) that is not well understood. In particular, unlike other IBDs, the inflamed parts of the intestine compromise deep layers of the tissue and are not continuous but separated and distributed through the whole gastrointestinal tract, displaying a patchy inflammatory pattern. In the present paper, we introduce a toy-model which might explain the appearance of such patterns. We consider a reaction-diffusion system involving bacteria and phagocyte and prove that, under certain conditions, this system might reproduce an activator-inhibitor dynamic leading to the occurrence of Turing-type instabilities. In other words, we prove the existence of stable stationary solutions that are spatially periodic and do not vanish in time. We also propose a set of parameters for which the system exhibits such phenomena and compare it with realistic parameters found in the literature. This is the first time, as far as we know, that a Turing pattern is investigated in inflammatory models.

The conditions used to describe the presence of an immune disease are often represented by interaction graphs. These informative, but intricate structures are susceptible to perturbations at different levels. The mode in which that perturbation occurs is still of utmost importance in areas such as cell reprogramming and therapeutics models. In this sense, module identification can be useful to well characterise the global graph architecture. To help us with this identification, we perform topological overlap-related measures. Thanks to these measures, the location of highly disease-specific module regulators is possible. Such regulators can perturb other nodes, potentially causing the entire system to change behaviour or collapse. We provide a geometric framework explaining such situations in the context of inflammatory bowel diseases (IBD). IBD are severe chronic disorders of the gastrointestinal tract whose incidence is dramatically increasing worldwide. Our approach models different IBD status as Riemannian manifolds defined by the graph Laplacian of two high throughput proteome screenings. It also identifies module regulators as singularities within the manifolds (the so-called singular manifolds). Furthermore, it reinterprets the characteristic nonlinear dynamics of IBD as compensatory responses to perturbations on those singularities. Then, particular reconfigurations of the immune system could make the disease status move towards an innocuous target state.

We consider a nonlinear heat equation with a double source: |u|p−1u|u|^{p-1}u and |∇u|q|\\nabla u|^q. This equation has a double interest: in ecology, it was used by Souplet (1996) as a population dynamics model; in mathematics, it was introduced by Chipot and Weissler (1989) as an intermediate equation between the semilinear heat equation and the Hamilton-Jacobi equation. Further interest in this equation comes from its lack of variational structure. In this paper, we intend to see whether the standard blow-up dynamics known for the standard semilinear heat equation (with |u|p−1u|u|^{p-1}u as the only source) can be modified by the addition of the second source (|∇u|q|\\nabla u|^q). Here arises a nice critical phenomenon at blow-up: - when q>2p/(p+1)q>2p/(p+1), the second source is subcritical in size with respect to the first, and we recover the classicial blow-up profile known for the standard semilinear case; - when q=2p/(p+1)q=2p/(p+1), both terms have the same size, and only partial blow-up descriptions are available. In this paper, we focus on this case, and start from scratch to: - first, formally justify the occurrence of a new blow-up profile, which is different from the standard semilinear case; - second, to rigorously justify the existence of a solution obeying that profile, thanks to the constructive method introduced by Bricmont and Kupiainen together with Merle and Zaag. Note that our method yields the stability of the constructed solution. Moreover, our method is far from being a straightforward adaptation of earlier literature and should be considered as a source of novel ideas whose application goes beyond the particular equation we are considering, as we explain in the introduction.

Ulcerative colitis is a chronic disease characterized by bleeding and ulcers in the colon. Disease severity assessment via colonoscopy videos is time-consuming and only focuses on the most severe lesions. Automated detection methods enable fine-grained assessment but depend on the training set quality. To suit the local clinical setup, an internal training dataset containing only rough bounding box annotations around lesions was utilized. Following previous works, we propose to use linear models in suitable color spaces to detect lesions. We introduce an efficient sampling scheme for exploring the set of linear classifiers and removing trivial models i.e., those showing zero false negative or positive ratios. Bounding boxes lead to exaggerated false detection ratios due to mislabeled pixels, especially in the corners, resulting in decreased model accuracy. Therefore, we propose to evaluate the model sensitivity on the annotation level instead of the pixel level. Our sampling strategy can eliminate up to 25% of trivial models. Despite the limited quality of annotations, the detectors achieved better performance in comparison with the state-of-the-art methods. When tested on a small subset of endoscopic images, the best models exhibit low variability. However, the inter-patient model performance was variable suggesting that appearance normalization is critical in this context.

In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity. The exponent p is superlinear and less than$1+\\frac{4}{N-1}$if N ≥ 2.

This is the first of two papers devoted to the study of the properties of the blow-up surface for the NN dimensional semilinear wave equation with subconformal power nonlinearity. In a series of papers, we have clarified the situation in one space dimension. Our goal here is to extend some of the properties to higher dimension. In dimension one, an essential tool was to study the dynamics of the solution in similarity variables, near the set of non-zero equilibria, which are obtained by a Lorentz transform of the space-independent solution. As a matter of fact, the main part of this paper is to study similar objects in higher dimensions. More precisely, near that set of equilibria, we show that solutions are either non-global or go to zero or converge to some explicit equilibrium. We also show that the first case cannot occur in the characteristic case and that only the third possibility occurs in the non-characteristic case, thanks to the non-degeneracy of the blow-up limit, another new result in our paper. As a by-product of our techniques, we obtain the stability of the zero solution.

In this paper, we consider the complex Ginzburg-Landau equation ∂tu=(1+iβ)Δu+(1+iδ)|u|p-1u-αu,whereβ,δ,α∈R.The study focuses on investigating the finite-time blow-up phenomenon, which remains an open question for a broad range of parameters, particularly for β and δ. Specifically, for a fixed β∈R, the existence of finite-time blow-up solutions for arbitrarily large values of |δ| is still unknown. According to a conjecture made by Popp et al. (Physica D Nonlinear Phenom 114:81–107 1998), when β=0 and δ is large, blow-up does not occur for generic initial data. In this paper, we show that their conjecture is not valid for all types of initial data, by presenting the existence of blow-up solutions for β=0 and any δ∈R with different types of blowup.