Objective
The present project intends to develop a direct collaboration between institutes in Russia, Ukraine, Georgia, the University of Bath (UK) and the Paul Sabatier University (Toulouse, France).
The project lies in the area of nonlinear analysis dealing with nonlinear evolution partial differential equations. The emphasis is placed on the rigorous qualitative theory, with a strong basis in analysis, classical and functional. It also uses the machinery of ordinary differential equations and some geometrical and topological methods as well as numerical adaptive techniques. In particular, we consider the equations occurring as ground state equations in the field theory, as models for diffusive, convective, and/or reactive processes, as ignition paradigms or flame models in combustion. The published works of the participants show these applied aspects.
There are four subjects of common interest between the seven teams; namely,
- Nonlinear evolution equations and systems appearing in the description of a number of physical processes, mainly the thermal propagation, flows in porous media, and reaction-diffusion processes of different types.
- The stationary states, which usually take the form of solutions to nonlinear elliptic equations. These equations appear frequently as ground state equations in field equations or (rescaled) stationary profiles of self-similar processes.
- Singularities of different types arising in these nonlinear problems. The most important are: shocks in gas dynamics equations (called nonlinear conservation laws in the mathematical literature), free boundary layers (typical in fluid mechanics), blow-up (one of the main mathematical aspects of combustion theory), quenching and extinction phenomena (important in reaction dynamics).
- The exploitation of geometrical or group theoretical properties is a great help in the study of nonlinear problems. The use of self-similarity has been of permanent interest of the Russian and UK teams involved.
The main expected results in the above-mentioned and other fields are the following:
- Blow-up criteria (including the gradient catastrophe) for quasilinear parabolic and hyperbolic equations and inequalities, as well as for systems of such equations and inequalities, in the whole space, half-space, cone-like and exterior domains.
- The existence and multiplicity results for positive solutions of evolution equations in unbounded domains with nonlinear boundary conditions and for the corresponding stationary problems. The asymptotic behaviour at infinity of such solutions.
Topic(s)
Call for proposal
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BATH
United Kingdom