具有非线性边界条件的发展型p-Laplace方程（组）解的存在性和爆破性
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摘要
本文首先讨论了如下具有非线性源项的发展型p-Laplace方程组的初边值问题其中T > 0, q11,q12,q21,q22 0,a,λ> 0, p1,p2 > 1.所讨论问题包括局部源,非局部源,非线性扩散项以及非线性边界条件对上述方程组解爆破的临界指标的影响.
对于f(u,v),g(u,v)为非线性局部源(形如um11vm12, um21vm22)的情况,作者讨论了上述方程组弱解的性质.通过抛物正则化方法和弱比较原理,得到了该问题弱解的局部存在性,整体存在性和爆破的条件.
对于f(u,v),g(u,v)为非线性非局部源(形如的情况,作者也讨论了该方程组弱解的局部存在性与整体性质.
本文还讨论了如下同时具有局部化源和非局部源的双重退化抛物方程的初边值问题其中T > 0, a,b,λ,k,m,p,r,q > 0, x0∈[0,1].作者获得了该问题弱解的局部存在性,并且讨论了各种非线性项对解整体性质的影响.
In the real world, di?usion phenomena exists widely. The mathemat-ical models are partly reduced to the study of some parabolic equationsor systems. In chemistry, ?uid mechanics, theory of phase transitions, im-age processing, biological populations, as well as areas such as percolationtheory, have raised a number of parabolic equations with appropriate ini-tial and boundary conditions to describe the di?usion phenomena, In therecent decades, many scholars have made significant progress in studyingsuch models.
Usually, people use some linear equations to describe some di?usionmodels, but most of the real models should be described with nonlinearequations. Therefore, the parabolic equations used to describe these mod-els may have nonlinear terms, and also may be degenerate or singular. Al-though the nonlinear problems may accurately re?ect some actual phenom-ena, but they also cause some di?culties in studying these problems. Forexample, the nonlinear terms (di?usion term, source term, boundary term)will play a role in promoting or hindering in blowing-up of the solutions.
This paper mainly studies some problems on some systems of evolu-tional p-Laplacian equations and systems of doubly degenerate parabolic equation. The topics include the e?ect of the nonlinear boundary condi-tion, nonlinear local sources, nonlinear localized sources, nonlinear nonlocalsources and coupling among them in the critical exponents of solutions.This thesis consist of three chapters.
In the Introduction, we recall the background of the related topics andsummarize the main results of the present thesis.
In Chapter 1, we have studied two systems of evolutional p-Laplacianequations with local sources and the nonlinear boundary conditions. Thereare di?erent degrees of the complexity of the nonlinear properties in twotypes of local sources. We have overcome the di?culties caused by thenonlinear terms. With the use of the method of parabolic regularizationand comparison principle, we prove a necessary and su?cient condition onblow-up of the solutions. We may notice that from the final results, thepromoting role in blow-up of the solutions, caused by di?erent nonlinearterms, is also di?erent.
In the first part of Chapter 1, we consider the following evolutionalp-Laplacian system with nonlinear local sources and nonlinear boundaryconditions:
where T > 0, m1,m2,q1,q2 > 0, p1,p2 > 1. Since the system has non-linear local sources vm1,um2 and nonlinear boundary conditions vq1,uq2,it is in general di?cult to study the system. We regularize the problem (1), prove some estimates for the solutions of the regularized problem, andhence obtain the local existence of solutions. Then, we prove a weak com-parison principle and obtain some necessary and su-cient conditions on theglobal existence of all positive(weak) solutions by constructing a supersolu-tion(subsolution) of the system. Our main results are the following:
Theorem 2 All positive weak solutions of the system (1) exist globallyif and only if m1m2 1, q1q2p1p2 1, m1p2q2 1 and m2p1q1 1.
Theorem 3 All positive weak solutions of the system (1) blow up infinite time if and only if m1m2 > 1, q1q2p1p2 > 1, m1p2q2 > 1 or m2p1q1 > 1.
In the second part of Chapter 1, we study the evolutional p-Laplaciansystem with more complicated local sources:where T > 0, m11,m12,m21,m22,q1,q2 > 0, p1,p2 > 1. We also prove thelocal existence of the weak solution of the problem. And then, we discussthe global existence and blow-up of weak solutions. The main results arethe following:
Theorem 4 All positive weak solutions of the system (2) exist globallyif and only if m11m22 1, p1p2q1q2 1, m12m21 (1 - m11)(1 - m22),m21p1q1 1 - m22 and m12p2q2 1 - m11.
Theorem 5 All positive weak solutions of the system (2) blow up infinite time if and only if m11m22 > 1, p1p2q1q2 > 1, m12m21 > (1-m11)(1-m22), m21p1q1 > 1 - m22 or m12p2q2 > 1 - m11.
In Chapter 2, we study two systems of evolutional p-Laplacian equa-tions with nonlinear boundary conditions. The first system has nonlinearnonlocal sources. The second system has coupled local and nonlocal sources.Compared with the equations in Chapter 1, there are di-erent types of thenonlinear sources and complexity of the nonlinear boundary conditions ofthe equations in Chapter 2. In the second chapter, for the source term andcomplexity of the boundary conditions, we conducted a more detailed dis-cussion and finally obtain some necessary and su-cient conditions to theexistence of the solutions.
In the first part of Chapter 2, we consider the following evolutionalp-Laplacian system with nonlinear nonlocal sources and more complicatedboundary conditions:where T > 0, m1,m2,q11,q12,q21,q22 > 0, p1,p2 > 1. We have overcomethe di-culties caused by the nonlocal sources and the boundary conditions,and obtain the local existence of the weak solutions to the system. Then wediscuss the global existence(blow-up) of weak solutions of the system. Themain results are the following theorems:
Theorem 6 All positive weak solutions of the system (3) exist globallyif and only if m1m2 1, q12q21p1p2 (1 - p2q22)(1 - p1q11), m1p2q211 - p2q22 and m2p1q12 1 - p1q11.
Theorem 7 All positive weak solutions of the system (3) blow up infinite time if one of the following inequalities holds:
i) m1m2 > 1;
ii) m1m2 1, q12q21p1p2 (1-p2q22)(1-p1q11) and m1p2q21 > 1-p2q22;
iii) m1m2 1, q12q21p1p2 (1 - p2q22)(1 - p1q11) and m2p1q12 >1 - p1q11;iv) m1m2 1, q12q21p1p2 > (1 - p2q22)(1 - p1q11).
Then, in the second part of Chapter 2, we consider the evolutionalp-Laplacian system with more complicated nonlocal sources:-where T > 0, m11,m12,m21,m22,q11,q12,q21,q22,a,b,λ> 0, p1,p2 > 1.With this type of source terms, we discuss the local existence, global exis-tence and blow-up of the weak solutions. The main results are the followingtheorems:
Theorem 8 All positive weak solutions of the system (4) exist globallyif and only if m11m21 1, (1 - m11)(1 - m21) m12m22, q12q21p1p2(1 - p2q22)(1 - p1q11), m12p2q21 (1 - m11)(1 - p2q22) and m22p1q12(1 - m21)(1 - p1q11).
Theorem 9 All positive weak solutions of the system (4) blow up infinite time if one of the following inequalities holds:
i) m11 > 1 and m21 > 1;
ii) m11m21 1 and (1 - m11)(1 - m21) < m12m21;
iii) m11m21 1, (1 - m11)(1 - m21) m12m22, q12q21p1p2 (1 -p2q22)(1 - p1q11) and m12p2q21 > (1 - m11)(1 - p2q22);
iv) m11m21 1, (1 - m11)(1 - m21) m12m22, q12q21p1p2 (1 -p2q22)(1 - p1q11) and m22p1q12 > (1 - m21)(1 - p1q11);
v) m11m21 1, (1 - m11)(1 - m21) m12m22 and q12q21p1p2 > (1 -p2q22)(1 - p1q11).
In Chapter 3, we consider the following system of doubly degenerateparabolic equation with nonlinear nonlocal source and nonlinear localizedsource:where t > 0, a,b,λ,k,m,p,r,q > 0, x0∈[0,1]. We obtain:
Theorem 10 If k < p or k < r, All positive weak solutions of thesystem (5) blow up.
Theorem 11 If k p,r and k m, then when q > mk, all positiveweak solutions of the system (5) blow up; when q mk, all positive weaksolutions of the system exist globally.
Theorem 12 If k p,r and k > m, then when q > all positiveweak solutions of the system (5) blow up; when q , all positive weaksolutions of the system exist globally.
对于f(u,v),g(u,v)为非线性局部源(形如um11vm12, um21vm22)的情况,作者讨论了上述方程组弱解的性质.通过抛物正则化方法和弱比较原理,得到了该问题弱解的局部存在性,整体存在性和爆破的条件.
对于f(u,v),g(u,v)为非线性非局部源(形如的情况,作者也讨论了该方程组弱解的局部存在性与整体性质.
本文还讨论了如下同时具有局部化源和非局部源的双重退化抛物方程的初边值问题其中T > 0, a,b,λ,k,m,p,r,q > 0, x0∈[0,1].作者获得了该问题弱解的局部存在性,并且讨论了各种非线性项对解整体性质的影响.
In the real world, di?usion phenomena exists widely. The mathemat-ical models are partly reduced to the study of some parabolic equationsor systems. In chemistry, ?uid mechanics, theory of phase transitions, im-age processing, biological populations, as well as areas such as percolationtheory, have raised a number of parabolic equations with appropriate ini-tial and boundary conditions to describe the di?usion phenomena, In therecent decades, many scholars have made significant progress in studyingsuch models.
Usually, people use some linear equations to describe some di?usionmodels, but most of the real models should be described with nonlinearequations. Therefore, the parabolic equations used to describe these mod-els may have nonlinear terms, and also may be degenerate or singular. Al-though the nonlinear problems may accurately re?ect some actual phenom-ena, but they also cause some di?culties in studying these problems. Forexample, the nonlinear terms (di?usion term, source term, boundary term)will play a role in promoting or hindering in blowing-up of the solutions.
This paper mainly studies some problems on some systems of evolu-tional p-Laplacian equations and systems of doubly degenerate parabolic equation. The topics include the e?ect of the nonlinear boundary condi-tion, nonlinear local sources, nonlinear localized sources, nonlinear nonlocalsources and coupling among them in the critical exponents of solutions.This thesis consist of three chapters.
In the Introduction, we recall the background of the related topics andsummarize the main results of the present thesis.
In Chapter 1, we have studied two systems of evolutional p-Laplacianequations with local sources and the nonlinear boundary conditions. Thereare di?erent degrees of the complexity of the nonlinear properties in twotypes of local sources. We have overcome the di?culties caused by thenonlinear terms. With the use of the method of parabolic regularizationand comparison principle, we prove a necessary and su?cient condition onblow-up of the solutions. We may notice that from the final results, thepromoting role in blow-up of the solutions, caused by di?erent nonlinearterms, is also di?erent.
In the first part of Chapter 1, we consider the following evolutionalp-Laplacian system with nonlinear local sources and nonlinear boundaryconditions:
where T > 0, m1,m2,q1,q2 > 0, p1,p2 > 1. Since the system has non-linear local sources vm1,um2 and nonlinear boundary conditions vq1,uq2,it is in general di?cult to study the system. We regularize the problem (1), prove some estimates for the solutions of the regularized problem, andhence obtain the local existence of solutions. Then, we prove a weak com-parison principle and obtain some necessary and su-cient conditions on theglobal existence of all positive(weak) solutions by constructing a supersolu-tion(subsolution) of the system. Our main results are the following:
Theorem 2 All positive weak solutions of the system (1) exist globallyif and only if m1m2 1, q1q2p1p2 1, m1p2q2 1 and m2p1q1 1.
Theorem 3 All positive weak solutions of the system (1) blow up infinite time if and only if m1m2 > 1, q1q2p1p2 > 1, m1p2q2 > 1 or m2p1q1 > 1.
In the second part of Chapter 1, we study the evolutional p-Laplaciansystem with more complicated local sources:where T > 0, m11,m12,m21,m22,q1,q2 > 0, p1,p2 > 1. We also prove thelocal existence of the weak solution of the problem. And then, we discussthe global existence and blow-up of weak solutions. The main results arethe following:
Theorem 4 All positive weak solutions of the system (2) exist globallyif and only if m11m22 1, p1p2q1q2 1, m12m21 (1 - m11)(1 - m22),m21p1q1 1 - m22 and m12p2q2 1 - m11.
Theorem 5 All positive weak solutions of the system (2) blow up infinite time if and only if m11m22 > 1, p1p2q1q2 > 1, m12m21 > (1-m11)(1-m22), m21p1q1 > 1 - m22 or m12p2q2 > 1 - m11.
In Chapter 2, we study two systems of evolutional p-Laplacian equa-tions with nonlinear boundary conditions. The first system has nonlinearnonlocal sources. The second system has coupled local and nonlocal sources.Compared with the equations in Chapter 1, there are di-erent types of thenonlinear sources and complexity of the nonlinear boundary conditions ofthe equations in Chapter 2. In the second chapter, for the source term andcomplexity of the boundary conditions, we conducted a more detailed dis-cussion and finally obtain some necessary and su-cient conditions to theexistence of the solutions.
In the first part of Chapter 2, we consider the following evolutionalp-Laplacian system with nonlinear nonlocal sources and more complicatedboundary conditions:where T > 0, m1,m2,q11,q12,q21,q22 > 0, p1,p2 > 1. We have overcomethe di-culties caused by the nonlocal sources and the boundary conditions,and obtain the local existence of the weak solutions to the system. Then wediscuss the global existence(blow-up) of weak solutions of the system. Themain results are the following theorems:
Theorem 6 All positive weak solutions of the system (3) exist globallyif and only if m1m2 1, q12q21p1p2 (1 - p2q22)(1 - p1q11), m1p2q211 - p2q22 and m2p1q12 1 - p1q11.
Theorem 7 All positive weak solutions of the system (3) blow up infinite time if one of the following inequalities holds:
i) m1m2 > 1;
ii) m1m2 1, q12q21p1p2 (1-p2q22)(1-p1q11) and m1p2q21 > 1-p2q22;
iii) m1m2 1, q12q21p1p2 (1 - p2q22)(1 - p1q11) and m2p1q12 >1 - p1q11;iv) m1m2 1, q12q21p1p2 > (1 - p2q22)(1 - p1q11).
Then, in the second part of Chapter 2, we consider the evolutionalp-Laplacian system with more complicated nonlocal sources:-where T > 0, m11,m12,m21,m22,q11,q12,q21,q22,a,b,λ> 0, p1,p2 > 1.With this type of source terms, we discuss the local existence, global exis-tence and blow-up of the weak solutions. The main results are the followingtheorems:
Theorem 8 All positive weak solutions of the system (4) exist globallyif and only if m11m21 1, (1 - m11)(1 - m21) m12m22, q12q21p1p2(1 - p2q22)(1 - p1q11), m12p2q21 (1 - m11)(1 - p2q22) and m22p1q12(1 - m21)(1 - p1q11).
Theorem 9 All positive weak solutions of the system (4) blow up infinite time if one of the following inequalities holds:
i) m11 > 1 and m21 > 1;
ii) m11m21 1 and (1 - m11)(1 - m21) < m12m21;
iii) m11m21 1, (1 - m11)(1 - m21) m12m22, q12q21p1p2 (1 -p2q22)(1 - p1q11) and m12p2q21 > (1 - m11)(1 - p2q22);
iv) m11m21 1, (1 - m11)(1 - m21) m12m22, q12q21p1p2 (1 -p2q22)(1 - p1q11) and m22p1q12 > (1 - m21)(1 - p1q11);
v) m11m21 1, (1 - m11)(1 - m21) m12m22 and q12q21p1p2 > (1 -p2q22)(1 - p1q11).
In Chapter 3, we consider the following system of doubly degenerateparabolic equation with nonlinear nonlocal source and nonlinear localizedsource:where t > 0, a,b,λ,k,m,p,r,q > 0, x0∈[0,1]. We obtain:
Theorem 10 If k < p or k < r, All positive weak solutions of thesystem (5) blow up.
Theorem 11 If k p,r and k m, then when q > mk, all positiveweak solutions of the system (5) blow up; when q mk, all positive weaksolutions of the system exist globally.
Theorem 12 If k p,r and k > m, then when q > all positiveweak solutions of the system (5) blow up; when q , all positive weaksolutions of the system exist globally.
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