This paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that each bounded positive solution of the convolution equation should either be asymptotically separated from zero or it should converge (exponentially) to zero. This dichotomy principle is then used to establish a general theorem guaranteeing the uniform persistence and existence of semi-wavefront solutions to the convolution equation. Finally, we apply our abstract results to several well-studied classes of evolution equations with asymmetric non-local and non-monotone response. We show that, contrary to the symmetric case, these equations can possess at the same time the stationary, the expansion and the extinction waves.Comment: 15 pages, submitte

We study the generalized bosonic string equation, in which f ∈ L 2 (R n ), then there exists a unique real-analytic solution to the Euclidean bosonic string in a Hilbert space H c,∞ (R n ) we define precisely below. Second, we consider the case in which the potential U (x, φ) in the generalized bosonic string equation depends nonlinearly on φ, and we show that this equation admits real-analytic solutions in H c,∞ (R n ) under some symmetry and growth assumptions on U . Finally, we show that the above given equation admits real-analytic solutions in H c,∞ (R n ) if U (x, φ) is suitably near U 0 (x, φ) = φ, even if no symmetry assumptions are imposed.

We study nonlocal linear equations of the form f(∂t)ϕ=J(t), t≥0, in which f is an entire function. We develop an appropriate functional calculus via Laplace transform, we solve the aforementioned equation completely in the space of exponentially bounded functions, and we analyze the delicate issue of the formulation of initial value problems for nonlocal equations.

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