Volume 6, Issue 1

In this article, we show the existence of a nontrivial solution of the quasilinear polyharmonic equation Δ n / m m u = f ( x , u ) in Ω, u = ∇ u = ⋯ = ∇ m -1 u = 0 on ∂Ω, where Ω is a bounded domain in ℝ n , n ≥ 2 m ≥ 2, and f behaves like e | u | n /( n - m ) as | u | → ∞.

A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock–Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let Θ t ( x ) = exp(- x 2 /(4 t ))/(4π t ) 1/2 be the heat kernel. With initial data f that is the distributional derivative of a continuous function, it is shown that u t ( x ) := u ( x , t ) := f * Θ t ( x ) is a classical solution of the heat equation u 11 = u 2 . The estimate ∥ f * Θ t ∥ ∞ ≤ ∥ f ∥/(π t ) 1/2 holds. The Alexiewicz norm is ∥ f ∥ := sup I |∫ I f |, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, ∥ u t - f ∥ → 0 as t → 0 + . The solution of the heat equation is unique under the assumptions that ∥ u t ∥ is bounded and u t → f in the Alexiewicz norm for some integrable f . The heat equation is also considered with initial data that is the n th derivative of a continuous function and in weighted spaces such that ∫ -∞ ∞ f ( x )exp(- a x 2 ) d x exists for some a > 0. Similar results are obtained.

In this paper, a mathematical model for a solid avascular tumor growth under the effect of periodic therapy is studied. Necessary and sufficient conditions for the global stability of tumor free equilibrium are given. The conditions under which there exists a unique periodic solution to the model are determined and we also show that the unique periodic solution is global attractor of all other positive solutions.

A linear functional (form) v is called regular if there exists a sequence of polynomials { S n } n ≥0 with deg S n = n which is orthogonal with respect to v . The linear functional v ˜ = S 1 v is not regular. We study properties of the linear functional u satisfying u = λ v ˜ + δ a , where a ∈ ℂ and λ ∈ ℂ - {0}. Necessary and sufficient conditions are given for the regularity of the linear functional u . The corresponding tridiagonal matrices and associated polynomials are also studied. A study of the semiclassical character of the found families is done. We conclude by giving some examples.