Explore the vast range of titles available.

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations: The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time

We consider semilinear parabolic equations of the form

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over

In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.

The author considers semilinear parabolic equations of the form u_t=u_xx+f(u),\\quad x\\in \\mathbb R,t>0, where f a C^1 function. Assuming that 0 and \\gamma >0 are constant steady states, the author investigates the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x,0) are near \\gamma for x\\approx -\\infty and near 0 for x\\approx \\infty . If the steady states 0 and \\gamma are both stable, the main theorem shows that at large times, the graph of u(\\cdot ,t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). The author proves this result without requiring monotonicity of u(\\cdot ,0) or the nondegeneracy of zeros of f. The case when one or both of the steady states 0, \\gamma is unstable is considered as well. As a corollary to the author's theorems, he shows that all front-like solutions are quasiconvergent: their \\omega -limit sets with respect to the locally uniform convergence consist of steady states. In the author's proofs he employs phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories \\{(u(x,t),u_x(x,t)):x\\in \\mathbb R\\}, t>0, of the solutions in question.

This monograph looks at several trends in the investigation of singular solutions of nonlinear elliptic and parabolic equations. It discusses results on the existence and properties of weak and entropy solutions for elliptic second-order equations and some classes of fourth-order equations with L1-data and questions on the removability of singularities of solutions to elliptic and parabolic second-order equations in divergence form. It looks at localized and nonlocalized singularly peaking boundary regimes for different classes of quasilinear parabolic second- and high-order equations in divergence form. The book will be useful for researchers and post-graduate students that specialize in the field of the theory of partial differential equations and nonlinear analysis. Contents: Foreword Part I: Nonlinear elliptic equations with L^1-data Nonlinear elliptic equations of the second order with L^1-data Nonlinear equations of the fourth order with strengthened coercivity and L^1-data Part II: Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order Removability of singularities of the solutions of quasilinear elliptic equations Removability of singularities of the solutions of quasilinear parabolic equations Quasilinear elliptic equations with coefficients from the Kato class Part III: Boundary regimes with peaking for quasilinear parabolic equations Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations Method of functional inequalities in peaking regimes for parabolic equations of higher orders Nonlocalized regimes with singular peaking Appendix: Formulations and proofs of the auxiliary results Bibliography

We consider fractional operators of the form H s = ( ∂ t - div x ( A ( x , t ) ∇ x ) ) s , ( x , t ) ∈ R n × R , where s ∈ ( 0 , 1 ) and A = A ( x , t ) = { A i , j ( x , t ) } i , j = 1 n is an accretive, bounded, complex, measurable, n × n -dimensional matrix valued function. We study the fractional operators H s and their relation to the initial value problem ( λ 1 - 2 s u ′ ) ′ ( λ ) = λ 1 - 2 s H u ( λ ) , λ ∈ ( 0 , ∞ ) , u ( 0 ) = u , in R + × R n × R . Exploring the relation, and making the additional assumption that A = A ( x , t ) = { A i , j ( x , t ) } i , j = 1 n is real, we derive some local properties of solutions to the non-local Dirichlet problem H s u = ( ∂ t - div x ( A ( x , t ) ∇ x ) ) s u = 0 for ( x , t ) ∈ Ω × J , u = f for ( x , t ) ∈ R n + 1 \\ ( Ω × J ) . Our contribution is that we allow for non-symmetric and time-dependent coefficients.

This paper is devoted to investigating the optimal convergence order of a weak Galerkin finite element approximation to a second-order parabolic equation whose solution has lower regularity. In many applications, the solution of a second-order parabolic equation has only H 1 + s smoothness with 0 < s < 1 , and the numerical experiments show that the weak Galerkin approximate solution exhibits an optimal convergence order of 1 + s . However, the standard numerical analysis for weak Galerkin finite element method always requires that the exact solution should have at least H 2 smoothness. Our work fills the gap in the error analysis of weak Galerkin finite element method under lower regularity condition, where we prove the convergence order is of 1 + s . The main strategy of analysis is to introduce an H 2 -regular finite element approximation to discretize the spatial variables in variational equation, and then we analyze the error between this semi-discretized solution and the full discretized weak Galerkin solution. Finally, we present some numerical experiments to validate the theoretical analysis.

New results on the strong solvability in Sobolev spaces of the quasilinear Venttsel’ problem for parabolic equations with discontinuous leading coefficients are obtained.