Journal of Evolution Equations最新文献

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian (mathcal {L}) on the Heisenberg group (mathbb {H}^n) with power type nonlinearity (|u|^p) and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space (dot{H}^{-gamma }_{mathcal {L}}(mathbb {H}^n), gamma >0), on (mathbb {H}^n). In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent (p_{text {crit}}(Q, gamma )=1+frac{4}{Q+2gamma },) for (gamma in (0, frac{Q}{2})), where (Q:=2n+2) is the homogeneous dimension of (mathbb {H}^n). More precisely, we establish

• A global-in-time existence of small data Sobolev solutions of lower regularity for (p>p_{text {crit}}(Q, gamma )) in the energy evolution space;

• A finite time blow-up of weak solutions for (1<p<p_{text {crit}}(Q, gamma )) under certain conditions on the initial data by using the test function method.

Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.

We study the solvability of the second boundary value problem of the Lagrangian mean curvature equation arising from special Lagrangian geometry. By the parabolic method, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle–Warren’s theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.

In this article, we prove that small localized data yield solutions to Kawahara-type equations which have linear dispersive decay on a finite time scale depending on the size of the initial data. We use the similar method used by Ifrim and Tataru to derive the dispersive decay bound of the solutions to the KdV equation, with some steps being simpler. This result is expected to be the first result of the small data global bounds of the fifth-order dispersive equations with quadratic nonlinearity.

We prove a compactness result related to G-convergence for autonomous evolutionary equations in the sense of Picard. Compared to previous work related to applications, we do not require any boundedness or regularity of the underlying spatial domain; nor do we assume any periodicity or ergodicity assumption on the potentially oscillatory part. In terms of abstract evolutionary equations, we remove any compactness assumptions of the resolvent modulo kernel of the spatial operator. To achieve the results, we introduced a slightly more general class of material laws. As a by-product, we also provide a criterion for G-convergence for time-dependent equations solely in terms of static equations.

We consider a non-homogeneous parabolic equation with degenerate coefficients of the form (u_t-L_{omega } u=u^p), where (L_{omega }=omega ^{-1}mathrm div(omega nabla )). This paper establishes the existence/non-existence of global-in-time mild solutions based on a critical exponent, known as the Fujita exponent. Similar topics for a semilinear heat equation with degenerate coefficients are treated in Fujishima (Calc Var Partial Differ Equ 58:25, 2019). They considered an equation (u_t-textrm{div}(omega nabla u) =u^p), which is not self-adjoint, with two types of homogeneous weights: (omega (x) = |x_1|^a) and (omega (x) = |x|^b) where (a,b>0). In this paper we consider the case of a self-adjoint operator, and extend to more general weights that meet certain restrictions such as being in the Muckenhoupt class (A_2), non-decreasing, and where the limits (alpha :=lim _{|x'|rightarrow infty }(log omega (x))/(log |x'|)) and (beta :=lim _{|x'|rightarrow 0}(log omega (x))/(log |x'|)) exist, where (x' = (x_1, dots , x_n)) and (1le nle N). The main result establishes that the Fujita exponent is given by (p_F = 1+2/(N+alpha )). This means that the asymptotic behavior of the weight at infinity affects global existence of solutions and the one at the origin does not.

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error

where (p in [2,infty )), U is the mild solution, (U^j) is obtained from a time discretisation scheme, k is the step size, and (N_k = T/k) for final time (T>0). This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error

which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

This paper is devoted to obtaining new lower bounds to the radius of spatial analyticity for the solutions of modified Korteweg–de Vries (mKdV) equation and a coupled system of mKdV-type equations, starting with real analytic initial data with a fixed radius of analyticity (sigma _0). Specifically, we derive almost conserved quantities to prove that the local solution can be extended to a time interval [0, T] for any large (T>0) in such a way that the radius of analyticity (sigma (T)) decays no faster than (cT^{-1}) for both the equations, where c is a positive constant. The results of this paper improve the ones obtained in Figueira and Panthee (Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion, to appear in NoDEA, arXiv:2307.09096) and Figueira and Himonas (J Math Anal Appl 497(2):124917, 2021), respectively, for the mKdV equation and a mKdV-type system.

The paper deals with three evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, but its derivation from the physical model is mathematically not fully satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces.

This paper concerns second-order sufficient conditions of optimality for a class of infinite dimensional optimal control problems under control constraints and end-point constraints. The distinctive feature of our formulation is the use of variational analysis techniques. Our approach is based on a suitable decomposition of controls, using second-order tangents, and it is developed in the case of functional constraints. An adaptation of the tools in the case of pointwise constraints on the controls is proposed too. We provide applications to concrete optimal control problems involving PDEs.

Let (0<theta le 2), (Nge 1) and (T>0). We are concerned with the Cauchy problem for the fractional semilinear parabolic equation

Here, (fin C[0,infty )) denotes a rather general growing nonlinearity and (u_0) may be unbounded. We study local in time solvability in the so-called critical and doubly critical cases. In particular, when (f(u)=u^{1+{theta }/{N}}left[ log (u+e)right] ^{a}), we obtain a sharp integrability condition on (u_0) which explicitly determines local in time existence/nonexistence of a nonnegative solution.