Maximum Principle Elliptic Equation

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Maximum Principle Elliptic Equation
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Finite Speed in Heat Transport
From image below and maximum principle one can
conclude that Heat Flux, propagating towards a
spot of snow should be melted, without increase
in temperature over whole area
It can be a different explanation of this
phenomena from mathematical point of view ( HW
give your explanation)
Explanation given by Luikov, based on
generalizing Heat transfer model by adding term
responsible for inertia effect
In the phenomenological heat conduction theory,
the heat distribution velocity is assumed
infinitely large. However in some medias it
should be taken into account that the heat
distribution velocity is not extremely high, but
has a certain finite value
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More about Heat Transformation
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Problem 1 Find distribution of the temperature in
the bar, of the length L with isolated side
surface, if temperature of the both ends
maintain equal zero for all time, and initial
function are 1) and 2)
Problem 2 Find distribution of the temperature in
the bar, of the length L with isolated side
surface, if both ends of the bar isolated for
all time, and initial function are 1) and 2)
Problem 3 Find distribution of the temperature in
the bar, of the length L with isolated side
surface, if one end of the bar isolated and on
another End temperature maintains zero all time,
and initial function are 1) and 2)
Problem 4 Find distribution of the temperature in
the bar, of the length L with isolated side
surface, if one of the end isolated and on
another there is a Constant heat flax equal Q
all time, and initial function are 1) and 2)
Problem 5 1) Give a physical interpretation of
obtained result 2) What kind of impact of initial
data 1) and 2) has on a solution behavior on
infinity
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Differential Heat Conduction Equation
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Reminder Assumptions, under which equations (2-4)
have been derived
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Obviously u1, and u0 are two steady state
solutions of equation (6). It is not difficult
to see that u0 is unstable solution, and u1 is
stable solution of the equation (6), in the
following sense If we will linearized equation
(6) around a steady state solution u0, then for
any small exist such a perturbation of 0
such that perturbed solution will be smaller
than but solution of the linearized equation
with this initial data will exponentially
increase. In the same time if we will linearized
equation (6) around a steady state solution u1,
then for any perturbation which is smaller than
1, and the solution of the linearized equation
with this initial data will exponentially
decrease.
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Important mathematical and biophysical features
of the KPP ( Abbreviation KPP Kolmogorov,
Petroskii, Piskunov.) They studied a traveling
wave phenomena for 1-D equation of type, in
contrast to A linear diffusive equation.