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The aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

This paper deals with a two-species attraction–repulsion chemotaxis system u t = Δ u - ξ 1 ∇ · ( u ∇ v ) + χ 1 ∇ · ( u ∇ z ) + f 1 ( u , w ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , τ v t = Δ v + w - v , ( x , t ) ∈ Ω × ( 0 , ∞ ) , w t = Δ w - ξ 2 ∇ · ( w ∇ z ) + χ 2 ∇ · ( w ∇ v ) + f 2 ( u , w ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , τ z t = Δ z + u - z , ( x , t ) ∈ Ω × ( 0 , ∞ ) , under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω ⊆ R n , where τ ∈ { 0 , 1 } , ξ i , χ i > 0 and f i ( u , w ) ( i = 1 , 2 ) satisfy f 1 ( u , w ) = u ( a 0 - a 1 u - a 2 w + a 3 ∫ Ω u d x + a 4 ∫ Ω w d x ) , f 2 ( u , w ) = w ( b 0 - b 1 u - b 2 w + b 3 ∫ Ω u d x + b 4 ∫ Ω w d x ) with a i , b i > 0 ( i = 0 , 1 , 2 ) , a j , b j ∈ R ( j = 3 , 4 ) . It is proved that in any space dimension n ≥ 1 , the above system possesses a unique global and uniformly bounded classical solution regardless of τ = 0 or τ = 1 under some suitable assumptions. Moreover, by constructing Lyapunov functionals, we establish the globally asymptotic stabilization of coexistence and semi-coexistence steady states.

Cells encounter a diverse array of physical and chemical signals as they navigate their natural surroundings. However, their response to the simultaneous presence of multiple cues remains elusive. Particularly, the impact of topography alongside a chemotactic gradient on cell migratory behavior remains insufficiently explored. In this paper, we investigate the effects of topographical obstacles during chemotaxis. Our approach involves modifying the Keller-Segel model, incorporating a spatially dependent coefficient of chemotaxis. Through our analysis, we demonstrate that this coefficient plays a crucial role in preventing blow-up phenomena in cell concentration.

We consider the Cauchy problem for a time fractional semilinear heat equation \"Equation missing\" where$$0<\\alpha <1,\\, \\gamma >1,\\, Nın \\mathbb {Z}_{\\geqslant 1}$$0 < α < 1 , γ > 1 , N ∈ Z ⩾ 1 and$$\\mu (x)$$μ ( x ) belongs to inhomogeneous/homogeneous Besov–Morrey spaces. The fractional derivative$$^{C}\\partial ^{\\alpha }_{t}$$C ∂ t α is interpreted in the Caputo sense. We present sufficient conditions for the existence of local/global-in-time solutions to problem (P). Our results cover all existing results in the literature and can be applied to a large class of initial data.

We show the finite time blow up of a solution to the Cauchy problem of a drift-diffusion equation of a parabolic-elliptic type in higher space dimensions. If the initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not exist globally in time and blows up in a finite time for the scaling critical space. Besides there exists a concentration point such that the solution exhibits the concentration in the critical norm. This type of blow up was observed in the scaling critical two dimensions. The proof is based on the profile decomposition and the Shannon inequality in the weighted space.

In the paper we prove the weighted Hardy type inequality 1 ∫ R N V φ 2 μ ( x ) d x ≤ ∫ R N | ∇ φ | 2 μ ( x ) d x + K ∫ R N φ 2 μ ( x ) d x , for functions φ in a weighted Sobolev space H μ 1 , for a wider class of potentials V than inverse square potentials and for weight functions μ of a quite general type. The case μ = 1 is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators L u = Δ u + ∇ μ μ · ∇ u perturbed by singular potentials.

In this article, we introduce the parametrix technique in order to construct fundamental solutions as a general method based on semigroups and their generators. This leads to a probabilistic interpretation of the parametrix method that is amenable to Monte Carlo simulation. We consider the explicit examples of continuous diffusions and jump driven stochastic differential equations with Hölder continuous coefficients.

In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity for or , where is the tangential angle at the point on evolving curves. For , we show the flow exists globally and the rescaled flow has a full-time convergence. For or , we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.

We study the behavior of solutions of the Cauchy problem for a semilinear heat equation with critical nonlinearity in the sense of Joseph and Lundgren. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its optimal and sharp convergence rate of solutions with a critical exponent and two exponentially approaching initial data. This rate contains a logarithmic term which does not contain in the super critical nonlinearity case. Proofs are given by a comparison method based on matched asymptotic expansion.

We prove existence and uniqueness of solutions to a class of porous media equations driven by the fractional Laplacian when the initial data are positive finite Radon measures on the Euclidean space R d . For given solutions without a prescribed initial condition, the problem of existence and uniqueness of the initial trace is also addressed. By the same methods we can also treat weighted fractional porous media equations, with a weight that can be singular at the origin, and must have a sufficiently slow decay at infinity (power-like). In particular, we show that the Barenblatt-type solutions exist and are unique. Such a result has a crucial role in Grillo et al. (Discret Contin Dyn Syst 35:5927–5962, 2015 ), where the asymptotic behavior of solutions is investigated. Our uniqueness result solves a problem left open, even in the non-weighted case, in Vázquez (J Eur Math Soc 16:769–803, 2014 ).