Unbounded Solutions Research Articles

The Cauchy–Dirichlet problem for the fast diffusion equation on bounded domains

The Fast Diffusion Equation (FDE) ut=Δum, with m∈(0,1), is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy–Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc.In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case.

Spontaneous autophoretic motion of isotropic disks

It is theoretically known that an isotropic chemically active particle in an unbounded solution undergoes symmetry breaking when the intrinsic Péclet number ${{Pe}}$ exceeds a finite critical value (Michelin et al., Phys. Fluids, vol. 25, 2013, 061701). At that value, a transition takes place from a stationary state to spontaneous motion. In two dimensions, where no stationary state is possible in an unbounded domain, a linear stability analysis in a large bounded domain (Hu et al., Phys. Rev. Lett., vol. 123, 2019, 238004) reveals that the critical ${{Pe}}$ value slowly diminishes as the domain size increases. Motivated by these findings, we here consider an unbounded domain from the outset, addressing the two-dimensional problem of steady self-propulsion with a focus on the limit ${{Pe}}\ll 1$ . This singular limit is handled using matched asymptotic expansions, conceptually decomposing the fluid domain into a particle-scale region, where the leading-order solute transport is diffusive, and a remote region, where diffusion and advection are comparable. The expansion parameter is provided by the product of ${{Pe}}$ and $U$ , the unknown particle speed (normalised by the standard autophoretic scale). The problem is unconventional in that the scaling of $U$ with ${{Pe}}$ must be determined in the course of the perturbation analysis. The resulting approximation, $U=4\exp ({-2/{Pe}-\gamma _{E}-1})/{{Pe}}$ ( $\gamma _{E}$ being the Euler–Mascheroni constant), is in remarkable agreement with the numerical predictions of Hu et al. in the common interval of validity.

Unbounded Weak Solutions to Strongly q-Nonlinear Elliptic Systems. Local Regularity

Unbounded Weak Solutions to Strongly q-Nonlinear Elliptic Systems. Local Regularity

Wave dynamics of a viscoelastic liquid

A falling film of a weakly viscoelastic liquid (Walters’ liquid B′′) is studied in detail for low to moderate values of the Reynolds number. A linear stability analysis within the framework of the Orr–Sommerfeld-type boundary value problem reveals a destabilizing effect of the viscoelastic parameter near the threshold of instability but a stabilizing effect of the viscoelastic parameter on the primary instability far away from the threshold of instability. The surface equation is derived to decipher the existence of nonlinear waves based on the centre manifold approach. However, it renders an unbounded solution in the short-wave regime. In order to avoid singular behaviour found in the surface equation, a regularized equation, a simplified second-order model, and a full second-order model are proposed. In the linear regime, both the regularized equation and the model equations show excellent agreement with the results of the Orr–Sommerfeld-type boundary value problem, i.e., the viscoelastic parameter demonstrates a dual role in the primary instability. In the nonlinear regime, the results of the regularized equation are verified with those of the simplified second-order model. Both the regularized equation and the simplified second-order model show qualitatively similar characteristics of the nonlinear waves. The speed and maximum amplitude corresponding to the steady-state travelling waves attain fixed values in the drag-inertia regime. Moreover, the speed and maximum amplitude acquired from the regularized equation underestimate the results of the simplified second-order model when the Reynolds number lies in the drag-inertia regime. The time-dependent simulation of the regularized equation displays a train of solitary waves downstream to a monochromatic time periodic forcing at the inlet. The separation distance between the first two solitary pulses is computed, and it seems to approach an asymptotic value when the time is very large. This fact assures the possibility of a bound-state formation among the solitary pulses downstream for the weakly viscoelastic liquid.

English

english

Properties of given and detected unbounded solutions to a class of chemotaxis models

AbstractThis paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow‐up time are established. Finally, for a simplified version of the model, some blow‐up criteria are proved.More precisely, we analyze a zero‐flux chemotaxis system essentially described as The problem is formulated in a bounded and smooth domain Ω of , with , for some , , , and with . A sufficiently regular initial data is also fixed.Under specific relations involving the above parameters, one of these always requiring some largeness conditions on , we prove that any given solution to (), blowing up at some finite time becomes also unbounded in ‐norm, for all ; we give lower bounds T (depending on ) of for the aforementioned solutions in some ‐norm, being ; whenever , we establish sufficient conditions on the parameters ensuring that for some u0 solutions to () effectively are unbounded at some finite time. Within the context of blow‐up phenomena connected to problem (), this research partially improves the analysis in Wang et al. (J Math Anal Appl. 2023;518(1):126679) and, moreover, contributes to enrich the level of knowledge on the topic.

Qualitative properties of solutions for dual fractional nonlinear parabolic equations

In this paper, we consider the dual fractional parabolic problem{∂tαu(x,t)+(−Δ)su(x,t)=f(u(x,t)),inR+n×R,u(x,t)=0,in(Rn﹨R+n)×R, where R+n:={x∈Rn|x1>0} is the right half space. We prove that the positive solutions are strictly increasing in x1 direction without assuming the solutions be bounded.So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative ∂tα and the fractional Laplacian (−Δ)s. To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several new ideas and our approaches are quite different from those in the previous literature. We first establish an unbounded narrow region principle without imposing any decay and boundedness assumptions on the antisymmetric functions at infinity by estimating the nonlocal operator ∂tα+(−Δ)s along a sequence of suitable auxiliary functions at their minimum points, which is an essential ingredient to carry out the method of moving planes at the starting point. Then in order to remove the decay or boundedness assumption on the solutions, we develop a new novel approach lies in establishing the averaging effects for such nonlocal operator and apply these averaging effects twice to guarantee that the plane can be moved all the way to infinity to derive the monotonicity of solutions.We believe that the new ideas and techniques developed here will become very useful tools in studying the qualitative properties of solutions, in particular of those unbounded solutions, for a wide range of fractional elliptic and parabolic problems.

Avoiding critical mass phenomena by arbitrarily mild saturation of cross-diffusive fluxes in two-dimensional Keller-Segel-Navier-Stokes systems

The Keller-Segel-Navier-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(n)∇c),ct+u⋅∇c=Δc−c+n,ut+(u⋅∇)u=Δu+∇P+n∇Φ,∇⋅u=0,(⋆) is considered in a smoothly bounded domain Ω⊂R2, with a given function χ∈C2([0,∞)).It is shown that ifχ(n)→0as n→∞, then for all suitably regular initial data an associated no-flux/no-flux/Dirichlet problem for (⋆) possesses a globally defined and bounded classical solution.This particularly indicates that actually arbitrarily weak saturation of cross-diffusive fluxes at large population densities is sufficient to suppress critical mass phenomena with respect to the occurrence of unbounded solutions, as known to be present when χ≡const. even in the fluid-free case.

Existence and uniqueness of blow-up solution to a fully fractional thermostat model

In this article, we deal with a fully fractional thermostat model in the settings of the Riemann–Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra–Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray–Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions — the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions.

Weak Harnack inequality for unbounded solutions to the p(x)-Laplace equation under the precise non-logarithmic conditions

Weak Harnack inequality for unbounded solutions to the p(x)-Laplace equation under the precise non-logarithmic conditions

Synthesizing Formal Network Specifications From Input-Output Examples

We propose, a tool that synthesizes network specifications in a declarative logic programming language from input-output examples. aims to accelerate the adoption of formal verification in networking practice, by reducing the effort and expertise required to specify network models or properties. aims to be i) highly expressive, capable of synthesizing network specifications with complex semantics; ii) scalable, by virtue of using a novel best-first search algorithm to efficiently explore an unbounded solution space, and iii) robust, avoiding the need for exhaustive input-output examples by actively generating new examples. Our experiments demonstrate that can synthesize a wide range of specifications used in network verification, analysis, and implementations. Furthermore, improves upon existing approaches in terms of expressiveness, robustness to examples, and the quality of synthesized programs.

An Improved Method for Order Reduction of High Order Uncertain SISO Dynamic Systems by Affine Arithmetic

This article presents a refined algorithm using Modified Polynomial Differentiation (MPD) method through Affine Arithmetic (AA) to reduce the high order uncertain systems. This new algorithm is applicable for the reduction of Continuous SISO systems. Measurement errors due to variation in ambient conditions may result the system as an uncertain system unlike other methods in literature. This strategy, unlike previous ones found in the Iteration literature, can produce a reduced order model that is stable from a original high order uncertain system that is also stable. Applications of Affine Arithmetic steer clear of many of the drawbacks of Interval Arithmetic, including unbounded solutions in a number of important applications. Utilizing common numerical examples from interval literature, the proposed approach has been validated

Propagation of velocity moments and uniqueness for the magnetized Vlasov–Poisson system

We present two results regarding the three-dimensional Vlasov–Poisson system in the full space with an external magnetic field. First, we investigate the propagation of velocity moments for solutions to the system when the magnetic field is uniform and time-dependent. We combine the classical moment approach with an induction procedure depending on the cyclotron period This allows us to obtain, like in the unmagnetized case, the propagation of velocity moments of order k > 2 in the full space case and of order k > 3 in the periodic case. Second, this time taking a general magnetic field that depends on both time and position, we manage to extend a result by Miot [A uniqueness criterion for unbounded solutions to the Vlasov–Poisson system, 2016] regarding uniqueness for Vlasov–Poisson to the magnetized framework.

The existence of unbounded solutions of asymmetric oscillations in the degenerate resonant case

We prove the existence of unbounded solutions of the asymmetric oscillation in the case when each zero of the discriminative function is degenerate. This is the only case that has not been studied in the literature.

Bounded and unbounded positive solutions for singular φ-Laplacians coupled system on the half-line with first-order derivative dependence

Bounded and unbounded positive solutions for singular φ-Laplacians coupled system on the half-line with first-order derivative dependence

ON A DUAL CHARACTERIZATION OF THE ASYMPTOTIC CONE FOR THE SOLUTION SET OF A LINEAR OPTIMIZATION PROBLEM

The concept of the asymptotic cone is very useful in various branches of pure and applied mathematics, especially in optimization and variational inequalities. In recent years, many authors and researchers have studied asymptotic directions and asymptotically convergent algorithms for unbounded solution sets. In this paper, we consider the asymptotic cone of the solution set Ω of a linear optimization problem and investigate various results on its asymptotic cone, asymptotic regularity, the dual and polar cones of the asymptotic cone, the support function of the solution set, etc. Finally, we present a dual characterization of the asymptotic cone Ω∞ for the solution set of a linear optimization problem.

Uniqueness for fractional parabolic and elliptic equations with drift

We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of linear, nonlocal parabolic problems with a drift. More precisely, the problem is nonlocal due to the presence of the fractional Laplacian as diffusion operator. The drift term is driven by a smooth enough, possibly unbounded vector field $ b $ which satisfies a suitable growth condition in the set $ \{x\in {\mathbb{R}}^N:\langle b(x), x\rangle> 0\} $. In general, our uniqueness class includes unbounded solutions; in particular, we get uniqueness of bounded solutions. Furthermore, we show sharpness of the hypothesis on the drift term $ b $; in fact we show that, if the drift term $ b $ violates, in an appropriate sense, the mentioned growth condition (see (2.5)), then infinitely many bounded solutions to the problem exist. Finally, we also investigate uniqueness of a linear, nonlocal elliptic equation with a drift term obtaining similar results.

Concavity Method: A concise survey

This short review article discusses the concavity method, one of the most effective ways to deal with parabolic equations with unbounded solutions in finite time. If the solution ceases to exist for some time, we say it blows up. The solution or some of its derivatives become singular depending on the equation. We focus on situations where the solution becomes unbounded in finite time, and our objective is to review some of the key blowup theory papers utilising the concavity method.

Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs

This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables y and z) in a super-linear way like |y||ln|y||δ+|z||ln|z||λ for some δ∈[0,1] and λ≥0. Let k be the maximum of δ, λ+1/2 and 2λ. For the following four different ranges of the growth power parameter k: k=1/2, k∈(1/2,1), k=1 and k>1, we give reasonably weakest possible different integrability conditions on the terminal value for ensuring existence and uniqueness of the unbounded solution to the BSDE. In the first two cases, they are stronger than the LlnL-integrability and weaker than any Lp-integrability with p>1; in the third case, the integrability condition is just some Lp-integrability for p>1; and in the last case, the integrability condition is stronger than any Lp-integrability with p>1 and weaker than any exp(Lɛ)-integrability with ɛ∈(0,1). We also establish three comparison theorems, which yield immediately the uniqueness, when either one of generators of both BSDEs is convex (or concave) in both unknown variables (y,z), or satisfies a one-sided Osgood condition in the first unknown variable y and a uniform continuity condition in the second unknown variable z. Finally, we give an application of our results in mathematical finance.

On the Effects of Boundary Conditions in One-Dimensional Models of Hemodynamics

The paper is devoted to the theoretical analysis of the effects of boundary conditions on the solutions of the system of one-dimensional (1D) hemodynamics. The integral inequalities, which realize the energy inequalities for the solutions of initial-boundary-value problems, are obtained. It is demonstrated that the unphysical unbounded solutions can take place for the case of bounded functions from boundary conditions. For the periodic boundary conditions, the integral estimation illustrates the correct behavior of the solution. For this case of boundary conditions, the effective Fourier method for the analytical solution is proposed. The analytical solutions, obtained by this approach, can be used for the comparison of different 1D blood-flow models. The results obtained in the paper allow for an the alternatively view of the stated boundary conditions and can explain some problems, which can arise in numerical simulations. They expand the possibilities of the application of analytical methods in the field of blood-flow simulation. The results can be useful for the specialists on blood-flow modeling.