Journal of Evolution Equations最新文献

Generalized Feller theory provides an important analog to Feller theory beyond locally compact metric state spaces. This is very useful for solutions of certain stochastic partial differential equations, Markovian lifts of fractional processes, or infinite-dimensional affine and polynomial processes which appear prominently in the theory of signature stochastic differential equations. We extend several folklore results related to generalized Feller processes, in particular on their construction and path properties, and provide the often quite sophisticated proofs in full detail. We also introduce the new concept of extended Feller processes and compare them with classical and generalized ones as well as with Doob's h-transform. A key example relates generalized Feller semigroups of algebra homomorphisms via the method of characteristics to transport equations and continuous semiflows on weighted spaces, i.e., a remarkably generic way to treat differential equations on weighted spaces. We also provide a counterexample, which shows that no condition of the basic definition of generalized Feller semigroups can be dropped.

Chemotaxis phenomena govern the directed movement of microorganisms in response to chemical stimuli. In this paper, we investigate two Keller-Segel systems of reaction-advection-diffusion equations modeling chemotaxis on thin networks. The distinction between two systems is driven by the rate of diffusion of the chemo-attractant. The intermediate rate of diffusion is modeled by a coupled pair of parabolic equations, while the rapid rate is described by a parabolic equation coupled with an elliptic one. Assuming the polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove local well-posedness of both systems on compact metric graphs, and, in particular, prove existence of unique classical solutions. This is achieved by constructing sufficiently regular mild solutions via analytic semigroup methods and combinatorial description of the heat kernel on metric graphs. The regularity of mild solutions is shown by applying abstract semigroup results to semi-linear parabolic equations on compact graphs. In addition, for logistic-type Keller-Segel systems we prove global well-posedness and, in some special cases, global uniform boundedness of solutions.

We use the framework of the first-order differential structure in metric measure spaces introduced by Gigli to define a notion of weak solution to gradient flows of convex, lower semicontinuous and coercive functionals. We prove their existence and uniqueness and show that they are also variational solutions; in particular, this is an existence result for variational solutions. Then, we apply this technique in the case of a gradient flow of a functional with inhomogeneous growth.

The well-posedness of the initial-boundary value problem for higher-order quadratic nonlinear Schrödinger equations on the half-line is studied by utilizing the Fokas solution formula for the corresponding linear problem. Using this formula, linear estimates are derived in Bourgain spaces for initial data in spatial Sobolev spaces on the half-line and boundary data in temporal Sobolev spaces suggested by the time regularity of the linear initial value problem. Then, the needed bilinear estimates are derived and used for showing that the iteration map defined via the Fokas solution formula is a contraction in appropriate solution spaces. Finally, well-posedness is established for optimal Sobolev exponents in a way analogous to the case of the initial value problem on the whole line with solutions in classical Bourgain spaces.

In an infinite-dimensional separable Hilbert space X, we study the realizations of Ornstein–Uhlenbeck evolution operators (P_{s,t}) in the spaces (L^p(X,gamma _t)), ({gamma _t}_{tin mathbb {R}}) being a suitable evolution system of measures for (P_{s,t}). We prove hypercontractivity results, relying on suitable Log-Sobolev estimates. Among the examples, we consider the transition evolution operator associated with a non-autonomous stochastic parabolic PDE.

In this paper, we show a blowup criterion of solution for a parabolic equation with critical exponential source and arbitrary positive initial energy, which generalizes the blowup conclusions in reference (Ishiwata et al. in J Evol Equ 21:1677–1716, 2021) for subcritical and critical initial energy cases that depend on the depth of the potential well. Additionally, the continuous dependence of the local solution on the initial data is proved in detail.

We study a free boundary value problem modelling the motion of a piston in a viscous compressible fluid. The fluid is modelled by 1D compressible Navier–Stokes equations with possibly degenerate viscosity coefficient, and the motion of the piston is described by Newton’s second law. We show that the initial boundary value problem has a unique global in time solution, and we also determine the large time behaviour of the system. Finally, we show how our methodology may be adapted to the motion of several pistons.

In this paper, we focus on investigating the existence of mild solutions and asymptotically almost periodic mild solutions for a class of partial differential inclusions. These inclusions involve a forcing multivalued function that relies on implicit spatial derivatives of the state variable. We introduce a novel approach to simplify the complexities associated with singularities when taking the (alpha )-norm.

Motivated by the competition model of two species with nonautonomous diffusion, we consider fully nonautonomous parabolic evolution equation of the form (frac{textrm{d}u}{textrm{d}t} + A(t)u(t) = f(t,u)+g(t)) in which the time-dependent family of linear partial differential operator A(t), the nonlinear term f(t, u), and the external force g is 1-periodic with respect to t. We prove the existence and uniqueness of a periodic solution of the above equation and study the inertial manifold for the solutions nearby that solution. We prove the existence of such an inertial manifold in the cases that the family of linear partial differential operators ((A(t))_{tin mathbb {R}}) generates an evolution family ((U(t,s))_{tge s}) satisfying certain dichotomy estimates, and the nonlinear term f(t, x) satisfies the (varphi )-Lipschitz condition, i.e., (left| f(t,x_1)-f(t,x_2)right| leqslant varphi (t)left| A(t)^{theta } (x_1-x_2)right| ) where (varphi (cdot )) belongs to some admissible function space on the whole line. Then, we apply our abstract results to the above-mentioned competition model of two species with nonautonomous diffusion.