TK
T. Kamerling
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Will It Blow Up?
Existence of Global Solutions of Weighted Nonlocal Reaction-Diffusion Problems
In this thesis we study the existence of global classical solutions to a weighted nonlocal reaction-diffusion equation on the whole space R
m
. In this equation we consider the evolution as a result of diffusion, nonlinear local growth and a weighted nonlocal inhibition term that considers the current state of the entire domain. These equations are particularly useful for models in biology, chemistry and evolution theory.
The main objective is to extend an existing global existence result for an unweighted nonlocal reaction-diffusion equation to a weighted one for a positive bounded weight. Unlike the original result, showing only the boundedness of solutions, we show the entire process of proving the global existence of classical solutions.
After introducing the necessary analytical tools and notations we go about proving the main result in five steps. Firstly we prove the existence of a unique mild solution using the well-known Banach fixed-point method. Standard regularity results are then used to obtain a local classical solution. With a classical solution in hand we can study various properties of the solution. First, it is shown that non-negative initial data results in non-negative solutions by means of bounding the size of the negative part of the solution by 0. Next, uniform a priori estimates are derived in L
k
(R
m
) for β≤k≤∞, where β comes from the nonlinearity inside the nonlocal inhibition term. These estimates hold under some restrictions concerning the exponents of the equation and the dimension of the x-domain. The most important aspect of these estimates is that they are independent of the local existence time. The proof for these estimates follows that of the unweighted nonlocal reaction-diffusion equation.
Finally, a continuation argument is employed, using the a priori estimate and the fact that it is independent of the local existence time, to show that every unique local classical solution of an equation satisfying the restrictions needed for the a priori estimate can be extended to a unique global classical solution. This concludes the extension of the global existence of solutions to the weighted problem.
As a last note, we also discuss the possibility of further study into the existence of global solutions for less restricted weights. ...
m
. In this equation we consider the evolution as a result of diffusion, nonlinear local growth and a weighted nonlocal inhibition term that considers the current state of the entire domain. These equations are particularly useful for models in biology, chemistry and evolution theory.
The main objective is to extend an existing global existence result for an unweighted nonlocal reaction-diffusion equation to a weighted one for a positive bounded weight. Unlike the original result, showing only the boundedness of solutions, we show the entire process of proving the global existence of classical solutions.
After introducing the necessary analytical tools and notations we go about proving the main result in five steps. Firstly we prove the existence of a unique mild solution using the well-known Banach fixed-point method. Standard regularity results are then used to obtain a local classical solution. With a classical solution in hand we can study various properties of the solution. First, it is shown that non-negative initial data results in non-negative solutions by means of bounding the size of the negative part of the solution by 0. Next, uniform a priori estimates are derived in L
k
(R
m
) for β≤k≤∞, where β comes from the nonlinearity inside the nonlocal inhibition term. These estimates hold under some restrictions concerning the exponents of the equation and the dimension of the x-domain. The most important aspect of these estimates is that they are independent of the local existence time. The proof for these estimates follows that of the unweighted nonlocal reaction-diffusion equation.
Finally, a continuation argument is employed, using the a priori estimate and the fact that it is independent of the local existence time, to show that every unique local classical solution of an equation satisfying the restrictions needed for the a priori estimate can be extended to a unique global classical solution. This concludes the extension of the global existence of solutions to the weighted problem.
As a last note, we also discuss the possibility of further study into the existence of global solutions for less restricted weights. ...
In this thesis we study the existence of global classical solutions to a weighted nonlocal reaction-diffusion equation on the whole space R
m
. In this equation we consider the evolution as a result of diffusion, nonlinear local growth and a weighted nonlocal inhibition term that considers the current state of the entire domain. These equations are particularly useful for models in biology, chemistry and evolution theory.
The main objective is to extend an existing global existence result for an unweighted nonlocal reaction-diffusion equation to a weighted one for a positive bounded weight. Unlike the original result, showing only the boundedness of solutions, we show the entire process of proving the global existence of classical solutions.
After introducing the necessary analytical tools and notations we go about proving the main result in five steps. Firstly we prove the existence of a unique mild solution using the well-known Banach fixed-point method. Standard regularity results are then used to obtain a local classical solution. With a classical solution in hand we can study various properties of the solution. First, it is shown that non-negative initial data results in non-negative solutions by means of bounding the size of the negative part of the solution by 0. Next, uniform a priori estimates are derived in L
k
(R
m
) for β≤k≤∞, where β comes from the nonlinearity inside the nonlocal inhibition term. These estimates hold under some restrictions concerning the exponents of the equation and the dimension of the x-domain. The most important aspect of these estimates is that they are independent of the local existence time. The proof for these estimates follows that of the unweighted nonlocal reaction-diffusion equation.
Finally, a continuation argument is employed, using the a priori estimate and the fact that it is independent of the local existence time, to show that every unique local classical solution of an equation satisfying the restrictions needed for the a priori estimate can be extended to a unique global classical solution. This concludes the extension of the global existence of solutions to the weighted problem.
As a last note, we also discuss the possibility of further study into the existence of global solutions for less restricted weights.
m
. In this equation we consider the evolution as a result of diffusion, nonlinear local growth and a weighted nonlocal inhibition term that considers the current state of the entire domain. These equations are particularly useful for models in biology, chemistry and evolution theory.
The main objective is to extend an existing global existence result for an unweighted nonlocal reaction-diffusion equation to a weighted one for a positive bounded weight. Unlike the original result, showing only the boundedness of solutions, we show the entire process of proving the global existence of classical solutions.
After introducing the necessary analytical tools and notations we go about proving the main result in five steps. Firstly we prove the existence of a unique mild solution using the well-known Banach fixed-point method. Standard regularity results are then used to obtain a local classical solution. With a classical solution in hand we can study various properties of the solution. First, it is shown that non-negative initial data results in non-negative solutions by means of bounding the size of the negative part of the solution by 0. Next, uniform a priori estimates are derived in L
k
(R
m
) for β≤k≤∞, where β comes from the nonlinearity inside the nonlocal inhibition term. These estimates hold under some restrictions concerning the exponents of the equation and the dimension of the x-domain. The most important aspect of these estimates is that they are independent of the local existence time. The proof for these estimates follows that of the unweighted nonlocal reaction-diffusion equation.
Finally, a continuation argument is employed, using the a priori estimate and the fact that it is independent of the local existence time, to show that every unique local classical solution of an equation satisfying the restrictions needed for the a priori estimate can be extended to a unique global classical solution. This concludes the extension of the global existence of solutions to the weighted problem.
As a last note, we also discuss the possibility of further study into the existence of global solutions for less restricted weights.