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Math 5440 Aaron Fogelson Fall, 2013 Math 5440 Problem Set 4 – Solutions 1: (Logan, 1.8 # 4) Find all radial solutions of the two-dimensional Laplace’s equation. That is, find all solutions of the form u(r)where r = p x2 +y2.Find the steady-state. Yesterday's America Solution! YOU are invited to take the responsibility to control your authority following the 6 key hierarchical principals.

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The Problem Of Blow-Up In Nonlinear Parabolic Equations

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Existence And Blow-Up For Higher-Order Semilinear Parabolic Equations: Majorizing Order-Preserving Operators

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'... . As a basic example, we establish that in the Cauchy problem for the 2m-th order semilinear parabolic equation u t = ( ) m u + juj p ; x 2 R N ; t > 0; u(x; 0) = u 0 (x); x 2 R N ; where m > 1, p > 1, with bounded integrable initial data u 0 , the critical Fujita exponent is pF = ...'
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. As a basic example, we establish that in the Cauchy problem for the 2m-th order semilinear parabolic equation u t = ( ) m u + juj p ; x 2 R N ; t > 0; u(x; 0) = u 0 (x); x 2 R N ; where m > 1, p > 1, with bounded integrable initial data u 0 , the critical Fujita exponent is pF = 1 + 2m=N , so that for p > pF there exists a class of small global solutions and for p 2 (1; pF ] blow-up can occur for arbitrarily small initial data. The analysis of the asymptotics of both classes of global and blow-up solutions is based on comparison with similarity solutions of the majorizing order-preserving equation, which is shown to exist for any m > 1. Generalizations of this idea to dierential and pseudodierential evolution equations and relations to positivity sets for higher-order equations are discussed. 1. Introduction: majorizing order-preserving operators and equations This paper deals with a class of higher-order semilinear parabolic dierential and nonlinear integral evolution...
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Comparison Results and Steady States for the Fujita Equation with Fractional Laplacian

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'... We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based ...'
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We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based on comparison with global solutions. For a critical power non-linearity we obtain a two-parameter family of radially symmetric stationary solutions. By extending

Blow-Up, Critical Exponents And Asymptotic Spectra For Nonlinear Hyperbolic Equations

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'... We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X , u tt = f 0 (u); t > 0; u(0) = u 0 ; u t (0) = u 1 ; where f : X ! R is a C¹-function. Several applications to the second and higher-order hyperbolic equations with local and nonloca ...'
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We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X , u tt = f 0 (u); t > 0; u(0) = u 0 ; u t (0) = u 1 ; where f : X ! R is a C¹-function. Several applications to the second and higher-order hyperbolic equations with local and nonlocal nonlinearities are presented. We also describe an approach to Kato's and John's critical exponents for the semilinear equations u t = u+b(x; t)juj p , p > 1, which are responsible for phenomena of stability, unstability, blow-up and asymptotic behaviour. We construct countable spectra of different asymptotic patterns of self-similar and non self-similar types for global and blow-up solutions for the autonomous equation u tt = u + juj p 1 u in different parameter ranges.
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Blow-Up Estimates for Higher-Order Semilinear Parabolic Equations

'... We prove L estimates on the blow-up behaviour of solutions of a 2m-th order semilinear parabolic equation u t = ( ) u + q(u); x 2 R ; t > 0; m > 1; with a general even function q(u) 0 with a superlinear growth for juj 1. Our comparison approach and estimates apply to general int ...'
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We prove L estimates on the blow-up behaviour of solutions of a 2m-th order semilinear parabolic equation u t = ( ) u + q(u); x 2 R ; t > 0; m > 1; with a general even function q(u) 0 with a superlinear growth for juj 1. Our comparison approach and estimates apply to general integral evolution equations. We also study the following problem: nd a continuous function q(u) with a superlinear growth as u !1 such that the parabolic equation exhibits regional blow-up in a domain of nite non-zero measure. We show that such a regional blow-up can occur for q(u) = uj ln jujj . We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t ! T is described by the self-similar solution U (x; t) = expf(T t) (x)g; : R ! C ; of the complex Hamilton-Jacobi equation U t = ( 1) 1 2m (rU rU)
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Fabrication TechnicianJob Locations US-VA-Wallops IslandID2020-2461# of Openings1CategoryOperationsOverview We are currently seeking candidates for a Fabrication Technician position, located in our Wallops Island, Virginia station. Surface Combat Systems Center (SCSC) Wallops Island, VA requires support for management, engineering, operation, and technical expertise for activation, operations, maintenance and engineering of equipment, systems and computer programs in support of all Naval Sea Systems Command (NAVSEA) and Program Executive Office, Integrated Warfare Systems (PEO IWS) missions and projects performed at SCSC. SCSC provides live and simulated integrated warfare capabilities in a net-centric, maritime environment to develop, test, evaluate, and conduct fleet operations and training for the warfighter. Responsibilities Work under the supervision of Engineering to complete and assist in the following duties: Provides comprehensive preventive and corrective maintenance, and troubleshooting support for the fiber and cable infrastructure at SCSC, including Aegis, Ship Self Defense System, and WIETC.Test the optical terminations and splices using a variety of specialty equipment for measuring optical power and associated optic losses.Testing and terminate CAT 5, CAT 6, multi pin and a variety of other connectors.Conduct all work in accordance with applicable industry and/or military cable standards.TravelThis position includes a low level of travel. Qualifications Education, Knowledge, Experience, Skills and Abilities Required:Five years experience working with fiber optics and commercial cable fabrication, inventory control and electric design and fabrication.Demonstrated ability to work both independently and in a team environment.Certificates, Licenses, Registrations:Current FF101 Certificate or equivalentCurrent ETA FO1 Certificate or equivalentQualifications:Must possess a valid Drivers LicenseMust have reliable transportation to and from various work locationsActive DoD Secret security clearance highly desired however candidates who are eligible to obtain and maintain a DoD Secret security clearance will be considered. If the selected candidate does not hold an active clearance, he/she will receive a contingent offer and start date will be delayed until an interim Secret clearance is granted.Physical Requirements:Ability to frequently traverse stairsSitting for long periods of time, standing, walking, crouching and kneelingTalking, hearing and seeingReaching, handling, using equipment, keyboards and mobile devicesLifting boxes (files and supplies) up to 20 lbsT-Solutions is an Equal Opportunity Employer. All qualified applicants will receive consideration for employment without regard to race, color, religion, sex, sexual orientation, gender identity, national origin, or protected veteran status and will not be discriminated against on the basis of disability.