Around the Field-Road Diffusion Model
Blow-up vs. Global Existence for a Fujita-Type Heat Exchanger System
During my thesis, I also focused on analyzing Fujita-type blow-up phenomena
in reaction-diffusion systems, particularly in a context of coupling unknowns through the diffusion mechanism.
To put the problem in context, in 1966, the mathematician
H.Fujita
considered the semi-linear equation
\( \partial_t u = \Delta u + u^{1+p} \)
in
\( \mathbb{R}^N \),
where he highlighted the existence of a critical exponent
\( p_F = 2/\! N \)
separating systematic blow-up
(\( p < p_F \))
from possible global existence
(\( p > p_F \))
of positive solutions.
This threshold, identified as \( 1/\! p = N/\! 2 \), corresponds to the balance ratio between the uniform algebraic decay rate of the Heat Equation
\[ \partial_t \mathbf{u} = \Delta \mathbf{u} \]
in
\( \Vert \mathbf{u} (t,\cdot) \Vert_{L^\infty(\mathbb{R}^N)} \sim C/t^{N/\! 2} \),
and the algebraic blow-up rate of the underlying ODE
\[ \frac{d}{dt}U = U^{1+p} \]
in
\( U(t) \sim C/(T_{boom}-t)^{1/\! p} \).
My work,
summarized in this article,
focused on the following Fujita-type system:
where the constant \( \kappa \) is either \( 0 \) or \( 1 \), acting as an on/off switch for the second non-linearity.
My first contribution was to analyze the associated linear problem, demonstrating that the solutions converge exponentially fast towards those of a certain decoupled parabolic system.
Subsequently, I determined the critical exponents \( p \) and \( q \) that separate systematic explosion from possible global existence.
Modeling the Field-Road System from a Stochastic Particle System
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