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This paper concerns the global in time existence of solutions for a semilinear heat equation (P) ∂ t u = Δ u + f (u) , x ∈ R N , t > 0 , u (x , 0) = u 0 (x) ≥ 0 , x ∈ R N , where N ≥ 1 , u 0 is a nonnegative initial function and f ∈ C 1 ([ 0 , ∞)) ∩ C 2 ((0 , ∞)) denotes superlinear nonlinearity of the problem. We consider the global in time existence and nonexistence of solutions for problem (P). The main purpose of this paper is to determine the critical decay rate of initial functions for the global existence of solutions. In particular, we show that it is characterized by quasi self-similar solutions which are solutions W of Δ W + y 2 ⋅ ∇ W + f (W) F (W) + f (W) + | ∇ W | 2 f (W) F (W) [ q − f ′ (W) F (W) ] = 0 in y ∈ R N , where F (s) ≔ ∫ s ∞ 1 f (η) d η and q ≥ 1. [ABSTRACT FROM AUTHOR]

Abstract Let Ω be a domain of R n , n ≥ 3. The classical Caffarelli–Kohn–Nirenberg inequality rewrites as the following inequality: for any s ∈ [ 0 , 2 ] and any γ < (n − 2) 2 4 , there exists a constant K (Ω , γ , s) > 0 such that (H S) ∫ Ω | u | 2 ⋆ (s) | x | s d x 2 2 ⋆ (s) ≤ K (Ω , γ , s) ∫ Ω | ∇ u | 2 − γ u 2 | x | 2 d x , for all u ∈ D 1 , 2 (Ω) (the completion of C c ∞ (Ω) for the relevant norm). When 0 ∈ Ω is an interior point, the range (− ∞ , (n − 2) 2 4) for γ cannot be improved: moreover, the optimal constant K (Ω , γ , s) is independent of Ω and there is no extremal for (H S). But when 0 ∈ ∂ Ω , the situation turns out to be drastically different since the geometry of the domain impacts : • the range of γ 's for which (H S) holds. • the value of the optimal constant K (Ω , γ , s) ; • the existence of extremals for (H S). When Ω is smooth, the problem was tackled by Ghoussoub–Robert (2017) where the role of the mean curvature was central. In the present paper, we consider nonsmooth domain with a singularity at 0 modeled on a cone. We show how the local geometry induced by the cone around the singularity influences the value of the Hardy constant on Ω. When γ is small, we introduce a new geometric object at the conical singularity that generalizes the "mean curvature": this allows to get extremals for (H S). The case of larger values for γ will be dealt in the forthcoming paper (Cheikh-Ali, 2018). As an intermediate result, we prove the symmetry of some solutions to singular pdes that has an interest on its own. [ABSTRACT FROM AUTHOR]

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation (− Δ) s u + μ u = (I α ∗ F (u)) f (u) on R N where s ∈ (0 , 1) , N ≥ 2 , α ∈ (0 , N) , μ > 0 , I α denotes the Riesz potential and F (t) = ∫ 0 t f (τ) d τ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than ∼ 1 | x | N + 2 s . The result is new even for homogeneous functions f (u) = | u | r − 2 u , r ∈ [ N + α N , 2) , and it complements the decays obtained in the linear and superlinear cases in Cingolani et al. (2022); D'Avenia et al. (2015). Differently from the local case s = 1 in Moroz and Van Schaftingen (2013), new phenomena arise connected to a new " s -sublinear" threshold that we detect on the growth of f. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers. [ABSTRACT FROM AUTHOR]

The objective of this paper is to deal with the deterministic problem consisting of non-linear heat equation of gradient type. The evolution equation emerges as projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree 2 n − 1 , onto the tangent space of a sphere M in a Hilbert space ℋ. We are going to deal with questions of the existence and the uniqueness of a global solution, and the invariance of the manifold, M i.e. if the suitable initial data lives on M then all trajectories of solutions also belong to M. Finally, we will also show the global solution generates a gradient flow. [ABSTRACT FROM AUTHOR]

We consider the periodic problem for the nonlinear damped wave equation with pumping and convective nonlinearity v t t + 2 α v t − v x x = v + ∂ x v x 3 , x ∈ Ω , t > 0 v (0 , x) = ϕ ˜ x , v t (0 , x) = ψ ˜ x , x ∈ Ω , where α > 0 , Ω = − π , π. We study the solutions, which satisfy the periodic boundary conditions v t , x = v t , 2 π + x for all x ∈ R and t > 0 , with the 2 π - periodic initial data ϕ ˜ x and ψ ˜ (x). Our aim in the present paper is to find the large time asymptotics for solutions to the periodic problem for the nonlinear damped wave equation (1.1) carefully studying the behavior of the first harmonics of the solution and applying the energy type estimates. We prove the following asymptotics for the solutions v t , x = ϕ ˜ ̂ 0 + 1 2 α ψ ˜ ̂ 0 + ɛ 1 + 2 b ɛ 2 t + O 1 + ɛ 2 t − 1 as t → ∞ uniformly with respect to x ∈ Ω , where ϕ ˜ ̂ 0 = ∫ Ω ϕ ˜ x d x , ψ ˜ ̂ 0 = ∫ Ω ψ ˜ x d x , ɛ = | ∫ Ω e i x ϕ ˜ x d x | , b = 3 2 α . [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness of negative radial solution to a Dirichlet problem of (m , k) -Hessian equation in a finite ball of R n for 1 < k < n m . Our proof is based on a Pohozaev identity and the monotone separation techniques. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear Choquard equation − Δ u + u = ln 1 | x | ∗ F (u) f (u) , in R 2 , where f ∈ C 1 (R , R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H 1 (R 2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11]. [ABSTRACT FROM AUTHOR]

We study a variant of the Fisher–KPP equation with nonlocal dispersal. Using the theory of large deviations, we show the emergence of a "Bramson-like" logarithmic delay for the linearised equation with step-like initial data. We conclude that the logarithmic delay emerges also for the solutions of the nonlinear equation. Previous papers found very precise results for the nonlinear equation with strong assumptions on the decay of the kernel. Our results are less precise, but they are valid for all thin-tailed kernels. [ABSTRACT FROM AUTHOR]

In this paper we provide a local well posedness result for a quasilinear beam-wave system of equations on a one-dimensional spatial domain under periodic and Dirichlet boundary conditions. This kind of systems provides a refined model for the time-evolution of suspension bridges, where the beam and wave equations describe respectively the longitudinal and torsional motion of the deck. The quasilinearity arises when one takes into account the nonlinear restoring action of deformable cables and hangers. To obtain the a priori estimates for the solutions of the linearized equation we build a modified energy by means of paradifferential changes of variables. Then we construct the solutions of the nonlinear problem by using a quasilinear iterative scheme à la Kato. [ABSTRACT FROM AUTHOR]

The Liouville type theorem on the parabolic Monge–Ampère equation − u t det D 2 u = 1 states that any entire parabolically convex classical solution must be of form − t + | x | 2 / 2 up to a re-scaling and transformation, under additional assumption that partial derivative with respect to time variable u t is strictly negative and bounded. In this paper, we study the case when u t is unbounded, prove an existence result of entire parabolically convex smooth solution and investigate the asymptotic behavior near infinity. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor in open manifolds. We prove the existence of such metrics by solving a class of fully nonlinear elliptic equations with the infinity Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a critical exponent with respect to the stability of solutions to an Hénon type equation on the hyperbolic space. In Euclidean space, there exists a critical exponent on stable solutions of an Hénon type equation. In particular, this exponent is called the Joseph–Lundgren exponent in case of the Lane–Emden equation. However, on the hyperbolic space, the Lane–Emden equation admits no such critical exponent on stability. We devote this paper to showing the existence of a critical exponent of Joseph–Lundgren type for a weighted Lane–Emden equation. More precisely, we prove non-existence of non-trivial stable solutions of the Hénon type equation for the subcritical case. Moreover, we show that the Hénon type equation has stable, positive, and radial solutions for the supercritical case. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the shallow water type equation u t + ∂ x 2 n + 1 u + 1 2 ∂ x (u 2) + ∂ x (1 − ∂ x 2) − 1 u 2 + 1 2 u x 2 = 0 with low regularity data in the periodic settings. Firstly, we proved that the bilinear estimate related to the nonlinear term of the equation in space W s (defined in page 5) is invalid with s < − n 2 + 1. Then, the locally well-posed of the Cauchy problem for the periodic shallow water-type equation is obtained in H s (T) with s > − n + 3 2 , n ≥ 2 for arbitrary initial data. Thus, our result improves the result of Himonas and Misiolek (Commun. Partial Differ. Equ, 23(1998), 123–139.), where they have proved that the problem is locally well-posed for small initial data in H s (T) with s ≥ − n 2 + 1 , n ∈ N + with the aid of the standard Fourier restriction norm method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exact asymptotic behavior of positive solution to the following singular boundary value problem Δ u = b (x) f (u) , x ∈ Ω , u > 0 in Ω , u | ∂ Ω = + ∞ , where Ω is a C 2 -bounded domain in R N , (N ≥ 3), f ∈ C 1 ((0 , ∞) , (0 , ∞)) is nondecreasing on (0 , ∞) and b is a function in C l o c γ (Ω) , (0 < γ < 1) such that there exist b 1 , b 2 > 0 satisfying for each x ∈ Ω , 0 < b 1 = lim d (x) ⟶ 0 inf b (x) h (d (x)) ≤ lim d (x) ⟶ 0 sup b (x) h (d (x)) = b 2 < ∞ , where d (x) = d i s t (x , ∂ Ω) and h (t) ≔ c t − λ exp ∫ t η z (s) s d s , η > d i a m (Ω) , λ ≤ 2 such that z is a continuous function on [ 0 , η ] with z (0) = 0. [ABSTRACT FROM AUTHOR]

Let (M n , g , e − ϕ d v) be a smooth metric measure space. In this paper, we derive a series of gradient estimates and a Harnack inequality for positive solutions of a nonlinear parabolic partial differential equation (Δ ϕ − ∂ t) u = q u + a u (ln u) α in M n × (− ∞ , + ∞) , where q ∈ C 2 , 1 ( M n × (− ∞ , + ∞)) and a , α ∈ R. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of elliptic equations in annular domains involving derivative terms. Using nonlinear analysis methods, some results regarding existence of solutions are established. More precisely, at least one radial solution is obtained basing on Schauder's fixed point theorem and contraction mapping theorem, respectively. Some iterative technique is also applied, which allows us to overcome difficulties that the nonlinearity is related to derivative terms. [ABSTRACT FROM AUTHOR]

In this paper, the existence and nonexistence of energy minimizer of the Kirchhoff–Schrödinger energy function with prescribed L 2 -norm in dimension four are considered. The energy infimum values are completely classified in terms of coefficient and exponent of the nonlinearity. The sharp existence results of global constraint minimizers for both the subcritical and critical exponent cases are obtained, and the criticality is in the sense of both Sobolev embedding and Gagliardo–Nirenberg inequality. Our results also show the delicate difference between the case without a trapping potential function and the one with potential function. [ABSTRACT FROM AUTHOR]

In this paper the existence and multiplicity of non-zero solutions for nonlinear Dirichlet problems involving the p -Laplacian operator and which are defined in the whole space is established. In particular, the existence of two non-zero solutions, one with negative energy and other with positive one for equations having combined effects of concave and convex nonlinearities is obtained. The approach is based on variational methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear Dirac–Bopp–Podolsky system − i ∑ k = 1 3 α k ∂ k u + [ V (x) + q ] β u + w u − ϕ u = f (x , u) , in R 3 , − △ ϕ + a 2 △ 2 ϕ = 4 π | u | 2 , in R 3 , where a , q > 0 , w ∈ R , V (x) is a potential function, and f (x , u) is the interaction term (nonlinearity). First, we give a physical motivation for this new kind of system. Second, under suitable assumptions on f and V , and by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of solutions (u , ϕ u). [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of W 2 , p (n < p < ∞) solutions of a double obstacle problem. Moreover, we show the optimal regularity of the solution and the local C 1 regularity of the free boundary under a thickness assumption at the free boundary point on the intersection of two free boundaries. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle ψ , F (D 2 u , x) = f χ Ω (u) ∩ { u < ψ } + F (D 2 ψ , x) χ Ω (u) ∩ { u = ψ } , u ≤ ψ in B 1 , where Ω (u) = B 1 ∖ { u = 0 } ∩ { ∇ u = 0 } . [ABSTRACT FROM AUTHOR]

In this paper, we study large-time behaviors for a fundamental model in nonlinear acoustics, precisely, the viscous Jordan–Moore–Gibson–Thompson (JMGT) equation in the whole space R n. This model describes nonlinear acoustics in perfect gases under irrotational flow and equipping Cattaneo's law of heat conduction. By employing refined WKB analysis and Fourier analysis, we derive first- and second-order asymptotic profiles of solution to the Moore–Gibson–Thompson (MGT) equation as t ≫ 1 , which illustrates novel optimal estimates for the solutions even subtracting its profiles. Concerning the nonlinear JMGT equation, via suggesting a new decomposition of nonlinear portion, we investigate the existence and large-time profiles of global (in time) small data Sobolev solutions with suitable regularity. These results help bridge a new connection between the JMGT equation and diffusion-waves as t ≫ 1. [ABSTRACT FROM AUTHOR]

This paper continues the development of regularity results for quasilinear elliptic equations − div (A (x , ∇ u)) = μ in Ω , and u = 0 on ∂ Ω , in Lorentz and Lorentz–Morrey spaces, where Ω ⊂ R n (n ≥ 2); A is a monotone Carathéodory vector valued operator acting between W 0 1 , p (Ω) and its dual W − 1 , p ′ (Ω) ; and μ is a datum in some Lebesgue space L m (Ω) , for m < p ′ . It emphasizes that in this paper, we restrict our study to the case of 'very singular' when 1 < p ≤ 3 n − 2 2 n − 1 , and under mild assumption that the p -capacity uniform thickness condition is imposed on the complement of domain Ω. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz–Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the 'very singular' case still remains a challenge, specifically related to Lorentz and Lorentz–Morrey spaces. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the Cauchy problem for a generalized nonlinear Schrödinger equation with spatial variable coefficients. A new weighted Sobolev space W 1 , q (R +) is introduced and the range of admissible pairs for the Strichartz estimates is improved. As an application, we show the local well-posedness result for the initial data in W 1 , q (R +). [ABSTRACT FROM AUTHOR]

Abstract This paper is focused on a one-dimensional nonlinear variational wave equation which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. We establish the local existence and uniqueness of classical solutions to its Cauchy problem with initial data given on the parabolic degenerating line. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior of viscosity solutions for a class of fully nonlinear second-order elliptic equations in the half space. Moreover, the result is applied to Monge-Ampère equations and consequently we improve some existent result. [ABSTRACT FROM AUTHOR]

In this paper we consider a semilinear parabolic equation in an infinite cylinder with shrinking cross section. The portion where the domain is not a straight half-cylinder is assumed to be compact. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the decreasing cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side. [ABSTRACT FROM AUTHOR]

We establish in this paper a priori global W 2 , δ estimates for singular fully nonlinear elliptic equations with L n right hand side terms. The method is to slide paraboloids and barrier functions vertically to touch the solution of the equation, and then to estimate the measure of the contact set in terms of the measure of the vertex point set. To derive global estimates from L n data, the Hardy–Littlewood maximal functions, appropriate localizations and a new type of covering argument are adopted. These methods also provide us a more direct proof of the W 2 , δ estimates for (nonsingular) fully nonlinear elliptic equations established by L. A. Caffarelli and X. Cabré. [ABSTRACT FROM AUTHOR]

In this paper we deal with a Dirichlet problem for an elliptic equation involving the 1-Laplacian operator and a source term. We prove that, when the growth of the source is subcritical, there exist two bounded nontrivial solutions to our problem. Moreover, a Pohoz̆aev type identity is proved, which holds even when the growth is supercritical. We also show explicit examples of our results. [ABSTRACT FROM AUTHOR]

In this paper we develop a global W 2 , p estimate for the viscosity solution of the Dirichlet problem of fully nonlinear elliptic equations F ( D 2 u , D u , u , x ) = f ( x ) in Ω , u = 0 on ∂ Ω to a more general function space. Given an N -function Φ and a Muckenhoupt weight w , we prove that if f belongs to the associated weighted Orlicz space L w Φ ( Ω ) , then D 2 u ∈ L w Φ ( Ω ) and u satisfies a global W w 2 , Φ estimate, under a minimal regularity requirement on F in the variable x and a basic geometric assumption on ∂ Ω . The correct condition on the couple, Φ and w , is also addressed. This result generalizes the W 2 , p estimate (Caffarelli, 1989, Escauriaza, 1993, Winter, 2009) of Calderón and Zygmund as well as an analogous one (Byun et al., 2016) in the weighted L p setting. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Evans–Krylov theorem for nonlocal parabolic fully nonlinear equations. [ABSTRACT FROM AUTHOR]

In this paper we study the nonlinear parabolic equation ∂ t u = Δ u + a | x | − γ | u | α u , t > 0 , x ∈ R N ∖ { 0 } , N ≥ 1 , a ∈ R , α > 0 , 0 < γ < min ( 2 , N ) and with initial value u ( 0 ) = φ . We establish local well-posedness in L q ( R N ) and in C 0 ( R N ) . In particular, the value q = N α / ( 2 − γ ) plays a critical role. For α > ( 2 − γ ) / N , we show the existence of global self-similar solutions with initial values φ ( x ) = ω ( x ) | x | − ( 2 − γ ) / α , where ω ∈ L ∞ ( R N ) is homogeneous of degree 0 and ‖ ω ‖ ∞ is sufficiently small. We then prove that if φ ( x ) ∼ ω ( x ) | x | − ( 2 − γ ) / α for | x | large, then the solution is global and is asymptotic in the L ∞ -norm to a self-similar solution of the nonlinear equation. While if φ ( x ) ∼ ω ( x ) | x | − σ for | x | large with ( 2 − γ ) / α < σ < N , then the solution is global but is asymptotic in the L ∞ -norm to e t Δ ( ω ( x ) | x | − σ ) . The equation with more general potential, ∂ t u = Δ u + V ( x ) | u | α u , V ( x ) | x | γ ∈ L ∞ ( R N ) , is also studied. In particular, for initial data φ ( x ) ∼ ω ( x ) | x | − ( 2 − γ ) / α , | x | large, we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity. [ABSTRACT FROM AUTHOR]

We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem. [ABSTRACT FROM AUTHOR]

In this paper, we establish gradient estimates for the positive bounded solutions to a general type of nonlinear parabolic equation concerning the weighted Laplacian ∂ ∂ t − a (x , t) − Δ f u (x , t) = F (u (x , t)) on a smooth metric measure space with the metric evolving under the (k , ∞) -super Perelman–Ricci flow and the Yamabe flow. Applications of our results include Liouville type results and gradient estimates for some important geometric partial differential equations such as the equations involving gradient Ricci solitons and the Einstein-scalar field Lichnerowicz type equations. Our results generalize and improve many previous works. [ABSTRACT FROM AUTHOR]

In this paper we use the modified method of potential wells to study the properties of solutions for a class of higher order nonlinear parabolic equations with p -Laplace term − ( | u x | p − 2 u x ) x and nonlocal source | u | q − 1 u − 1 ∣ Ω ∣ ∫ Ω | u | q − 1 u d x . Global existence, uniqueness, blow up in finite time and asymptotic behavior of solutions will be proved under different initial conditions. Furthermore, a numerical example is given to illustrate the blow-up of solutions in finite time. [ABSTRACT FROM AUTHOR]

This paper concerns with the existence of solitons, namely stable solitary waves, for the Benjamin–Ono and the fractional KdV equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear Dirac equation − i ε ∑ k = 1 3 α k ∂ k u + a β u + V (x) u = K (x) f (| u |) u , in R 3 , where ε > 0 is a small parameter, a > 0 is a constant, α 1 , α 2 , α 3 and β are 4 × 4 Pauli–Dirac matrices, V , K and f are continuous but are not necessarily of class C 1. We prove the existence of ground state solution by using variational methods, and we determine a concrete set related to the potentials V and K as the concentration position of these ground state solutions as ε → 0. Moreover, we consider some properties of these ground state solutions, such as convergence and exponential decay estimate. The result presented in this paper generalizes the result in Ding and Liu (2012). [ABSTRACT FROM AUTHOR]

In this paper, we identify the sharp energy threshold for the global existence and blow-up to the Hartree equation with a focusing and defocusing perturbation, respectively, by utilizing comprehensive potential-well structures generated by the combined nonlinearities. Then, by adopting the refined compactness lemma, we detect the dynamical properties of blow-up solutions near the sharp threshold mass, including blow-up rate, concentration, weak limits and limiting profile decomposition of blow-up solutions with small super critical mass. [ABSTRACT FROM AUTHOR]

This paper is devoted to the well-posedness of stochastic nonlinear Schrödinger equations in the energy space H 1 ( R d ) , and is a continuation of our recent work (Barbu et al., 2014). We allow general complex coefficients in the noise, thus covering both the conservative and non-conservative case, the latter being important to include quantum measurement effects in the underlying physical model. We consider both focusing and defocusing nonlinearities and prove global well-posedness in H 1 ( R d ) , including also the pathwise continuous dependence on initial conditions, with exponents of the nonlinearity in exactly the same (sharp) range as in the deterministic case. In particular, in the special conservative case, this work improves earlier results in de Bouard and Debussche (2003), where the critical exponents could not be reached for d ≥ 6 in the defocusing and for d ≥ 4 in the focusing case respectively. Moreover, the local existence, uniqueness and blowup alternative are also established for the energy-critical case. The approach presented here is mainly based on the rescaling approach already used in Barbu et al. (2014) to study the L 2 case and also on the Strichartz estimates established in Marzuola et al. (2008) for large perturbations of the Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we study monotone radially symmetric solutions of semilinear equations with Allen–Cahn type nonlinearities by the bifurcation method. Under suitable conditions imposed on the nonlinearities, we show that the structure of the monotone nodal solutions consists of a continuous U-shaped curve bifurcating from the trivial solution at the third eigenvalue of the Laplacian. The upper branch consists of a decreasing solution and the lower branch consists of an increasing solution. In particular, we show the following equation Δ u + λ ( u − u ∣ u ∣ p − 1 ) = 0 in B , ∂ u ∂ ν = 0 on ∂ B has exactly two monotone radial nodal solutions, one is decreasing and the other is increasing. Here B is the unit ball in R n , p > 1 and λ > 0 . [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove an interior Lorentz estimate of the spatial gradient of weak solutions to a class of nonlinear parabolic problems with an irregular obstacle in a bounded cylindrical domain when the nonlinearity is merely asymptotically regular. We assume that the associated nonlinearity satisfies certain ellipticity and growth conditions and the ( δ , R ) -vanishing condition. Here, the approach of constructing a regular obstacle problem via Poisson’s formula is used to establish Lorentz estimate for such an asymptotically regular obstacle problem. [ABSTRACT FROM AUTHOR]

In this paper we consider the semilinear elliptic problem { − Δ u = λ f ( u ) in Ω , u = 0 on ∂ Ω , where f is a nonnegative, locally Lipschitz continuous function, Ω is a smooth bounded domain and λ > 0 is a parameter. Under the assumption that f has an isolated positive zero α such that f ( t ) ( t − α ) N + 2 N − 2 is decreasing in ( α , α + δ ) , for some small δ > 0 , we show that for large enough λ there exist at least two positive solutions u λ < v λ , verifying ‖ u λ ‖ ∞ < α < ‖ v λ ‖ ∞ and u λ , v λ → α uniformly on compact subsets of Ω as λ → + ∞ . The existence of these solutions holds independently of the behavior of f near zero or infinity. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with regularity analysis for solutions to nonlinear partial differential equations. Many important practical problems are related with the p -Laplacian. Therefore, we are particularly interested in the smoothness of solutions to the p -Poisson equation. For the full range of parameters 1 < p < ∞ we investigate regularity estimates in the adaptivity scale B τ σ ( L τ ( Ω ) ) , 1 / τ = σ / d + 1 / p , of Besov spaces. The maximal smoothness σ in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to p -Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local Hölder with global Sobolev estimates. In particular, we prove that intersections of locally weighted Hölder spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper we study a measure, μ ˆ , associated with a positive p harmonic function u ˆ defined in an open set O ⊂ R n and vanishing on a portion Γ of ∂ O . If p > n we show μ ˆ is concentrated on a set of σ finite H n − 1 measure while if p = n the same conclusion holds provided Γ is uniformly fat in the sense of n capacity. Our work nearly answers in the affirmative a conjecture in Lewis (2015) and also appears to be the natural extension of Jones and Wolff (1988), Wolff (1993), to higher dimensions. [ABSTRACT FROM AUTHOR]

In [12], Feng and Nguyen (2012) define a cost on curves of measures which is finite exactly on the curves which solve a Fokker–Planck equation with L 2 drift. In this paper, using ideas of D. Gomes and E. Valdinoci, we give a different construction of the cost of Feng and Nguyen (2012). [ABSTRACT FROM AUTHOR]

Wave breaking of the water wave is important and interesting to physicists and mathematicians. In this paper, based on the conservation laws and the blow-up scenario, some new blow-up phenomena are derived for the Camassa–Holm equation and Degasperis–Procesi equation, which have attracted much attention due to their structure. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear viscoelastic equation: | u t | ρ u t t − △ u + ∫ 0 t g ( t − τ ) △ u ( τ ) d τ + | u t | m − 2 u t = | u | p − 2 u , in Ω × [ 0 , T ] , with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g , we prove the nonexistence of global solutions with positive initial energy. [ABSTRACT FROM AUTHOR]

We study the Dirichlet–Cauchy problem for nonlinear hyperbolic equation of second order in a domain with edges. The aim of this paper is to prove the regularity of solution in weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper we study the initial value problem of fully nonlinear third-order equations on the torus, that is ∂ t u = F ∂ x 3 u , ∂ x 2 u , ∂ x u , u , x , t with F a smooth function depending on the space variable x , the time variable t , the first three derivatives of u with respect to x , and u. In particular we find conditions on u (0) and F for which one can construct a local and unique solution u. In particular if F and u (0) satisfy some conditions then the equation behaves like a diffusive one and it has a parabolic smoothing property: the solution is infinitely smooth in one direction of time and the problem is ill-posed in the other direction of time. If F and u (0) satisfy some other conditions then the equation behaves like a dispersive one. We also prove continuous dependence with respect to the data. The proof relies upon energy estimates combined with a gauge transformation [6–8] and the Bona-Smith argument (Bona and Smith, 1975). [ABSTRACT FROM AUTHOR]

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the m -Laplacian operator − Δ m u = u q | ∇ u | p in R N , where N ⩾ 1 , m > 1 and p , q ⩾ 0. The technique of Bernstein gradient estimates is utilized to study the case p < m. Moreover, a Liouville-type theorem for supersolutions under subcritical range of exponents q (N − m) + p (N − 1) < N (m − 1) is also established. Then, we use a degree argument to obtain the existence of positive weak solutions for a nonlinear Dirichlet problem of the type − Δ m u = f (x , u , ∇ u) , with f satisfying certain structure conditions. Our proof is based on a priori estimates, which will be accomplished by using a blow-up argument together with the Liouville-type theorem in the half-space. As another application, some new Harnack inequalities are proved. [ABSTRACT FROM AUTHOR]

This paper is devoted to the optimal distributed control problem governed by a class of nonlinear dispersive equations with cubic nonlinearity, which contains the famous Novikov equation as special case. We first investigate the existence and uniqueness of the weak solution for the controlled system, and then find an optimal solution for the controlled system with the generalized cost functional. Moreover, by means of the method suggested by A.Ya. Dubovitskii and A.A. Milyutin, we establish the first order necessary optimality condition of optimal control for the controlled system in the fixed final horizon case. [ABSTRACT FROM AUTHOR]