Global Weak Solutions for the Weakly Dissipative Dullin-Gottwald-Holm Equation

In this paper, we are concerned with the existence and uniqueness of global weak solutions for the weakly dissipative Dullin-Gottwald-Holm equation describing the unidirectional propagation of surface waves in shallow water regime: ut − α2uxxt + c0ux + 3uux + γuxxx + λ(u − α2uxx) = α2(2uxuxx + uuxxx).Our main conclusion is that on c0 = − γ/α2 and λ ≥ 0, if the initial data satisfies certain sign conditions, then we show that the equation has corresponding strong solution which exists globally in time, finally we demonstrate the existence and uniqueness of global weak solutions to the equation.

Signalling Three-Way Intersections: Is Redundancy Better Than Only Mandatory or Prohibitory Signs?

At intersections, drivers need to infer which ways are allowed by interpreting mandatory and/or prohibitory traffic signs. Time and accuracy in this decision-making process are crucial factors to avoid accidents. Previous studies show that integrating information from prohibitory signs is generally more difficult than from mandatory signs. In Study 1, we compare combined redundant signalling conditions to simple sign conditions at three-way intersections. In Study 2, we carried out a survey among professionals responsible for signposting to test whether common practices are consistent with experimental research. In Study 1, an experimental task was applied (n=24), and in Study 2, the survey response rate was 17%. These included the main cities in Spain such as Madrid and Barcelona. Study 1 showed that inferences with mandatory signs are faster than those with prohibitory signs, and redundant information is an improvement only on prohibitory signs. In Study 2, prohibitory signs were those most frequently chosen by professionals responsible for signposting. In conclusion, the most used signs, according to the laboratory study, were not the best ones for signposting because the faster responses were obtained for mandatory signs, and in second place for redundant signs.

Strongly nonlinear elliptic unilateral problems without sign condition and with free obstacle in Musielak-Orlicz spaces

In this paper, we prove the existence of solutions to an elliptic problem containing two lower order terms, the first nonlinear term satisfying the growth conditions and without sign conditions and the second is a continuous function on R.

Nonexistence of Subcritical Solitary Waves

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to “half-solitary” waves (e.g. bores) which decay in only one direction.

Multivalued nonmonotone dynamic boundary condition

In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation

<p style='text-indent:20px;'>In this paper, we mainly study several problems on the weakly dissipative generalized Camassa-Holm equation. We first establish the local well-posedness of solutions by Kato's semigroup theory. We then derive the necessary and sufficient condition of the blow-up of solutions and a criteria to guarantee occurrence of wave breaking. Moreover, when the solution blows up, we obtain the precise blow-up rate. We finally show that the equation has a unique global solution provided the moment density associated with their initial datum satisfies appropriate sign conditions.</p>

Nonlinear nonlocal reaction-diffusion problem with local reaction

<p style='text-indent:20px;'>In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.</p>

Existence of Positive Weak Solutions for a New Class of p,q Laplacian Nonlinear Elliptic System with Sign-Changing Weights

In this paper, by using subsuper solutions method, we study the existence of weak positive solutions for a new class of p,q Laplacian nonlinear elliptic system in bounded domains, when ax, bx,αx, and βx are sign-changing functions that maybe negative near the boundary, without assuming sign conditions on f0,g0,h0, and γ0.

Earlier and More Robust Sensorimotor Discrimination of ASL Signs in Deaf Signers During Imitation

Background: Prior research suggests that the amount of experience an individual has with an action influences the degree to which the sensorimotor systems of their brain are involved in the subsequent perception of those actions. Less is known about how action experience and conceptual understanding impact sensorimotor involvement during imitation. We sought to explore this question by comparing a group of sign language users to a group of non-signers. We pitted the following two hypotheses against each other: 1) Deaf signers will show increased sensorimotor activity during sign imitation, and greater differentiation between sign types, due to greater prior experience and conceptual understanding of the signs; versus 2): Deaf signers will show less sensorimotor system activity and less differentiation of sign types in the sensorimotor system, because for those individuals sign imitation involves language systems of the brain more robustly than sensorimotor systems. We collected electroencephalograms (EEG) while the two groups imitated videos showing one-handed and two-handed ASL signs. Time-frequency data analysis was performed on alpha- and beta-range oscillations while they watched signs with the intent to imitate, and imitated the signs. During observation, deaf signers showed early differentiation in alpha/beta power between the one- and two-handed sign conditions, whereas hearing non-signers did not discriminate between the sign categories this way. Significant differences between groups were seen during sign imitation, wherein deaf signers showed desynchronization of alpha/beta EEG signals, and hearing non-signers showed increased power. The study suggests that in an imitative context, deaf signers engage anticipatory motor preparation in advance of action production, while hearing non-signers engage slower, more memory-related processes to help them complete with the complex task.