| Di?usion equations, as an important class of partial di?erential equations, comefrom a variety of di?usion phenomena appeared widely in nature. They are suggestedas mathematical models of physical problems in many ?elds such as ?ltration, imageprocessing, phase transition, biochemistry and dynamics of biological groups. Withthe intensive study by many researchers, di?usion equations have been thought of asan important branch of partial di?erential equations after developing in the past halfcenter. While, the research of nonclassical di?usion equations and classical di?usionequations add radiance and beauty to each other. In this monograph, we considerthe long time behavior of solutions of a class of non-classical di?usion equations—–pseudo-parabolic equations with nonlinear interior sources and the applications ofa class of hyperbolic—pseudo-parabolic equations to image processing. We mainlystudy the critical exponents of the Cauchy problems in the whole space Rn and thesecond initial-boundary value problems in the exterior space of the homogeneous andinhomogeneous semilinear pseudo-parabolic equations. After that, we use a relatedclass of mixed type equations: hyperbolic—pseudo-parabolic equations to image restoration. Nowadays, the majority researches on either the critical exponents orthe applications of partial di?erential equations to image processing concentrateon parabolic equations or hyperbolic equations, and few researches of the abovetwo aspects focus on the pseudo-parabolic equations, our researches will enrich thetheory and the applications to the engineering ?eld of partial di?erential equations.Pseudo-parabolic equation is an example of a general class of equations ofSobolev type, sometimes referred to as Sobolev-Galpern type. In the poineeringpaper of Sobolev [27] (1954), the equation for small osciallations in a rotation liquidwas obtained, it has the formIt is easy to see that, the highest-order term of the above equation has mixed timeand space derivatives. The equation with this character is said to be a Sobolevtype equation. The work of Sobolev ?re the interest of research on the non–classicaldi?usion equations. Gal'pern [49] studied the Cauchy problem for the equation ofthe formwhere M and L are linear elliptic operators; Showalter [51] investigated the ex-istence and uniqueness of strong solutions to the initial–boundary value problemwhen M and L are second order linear elliptic operators; Equations of the aboveform have been called pseudo-parabolic by Showalter and Ting [26], because well-posed initial-boundary value problems for parabolic equations are also well-posedfor the corresponding pseudo-parabolic equation. Moreover, in certain cases, thesolution of a parabolic initial-boundary value problem can be obtained as a limitof solutions to the corresponding problem for the pseudo-parabolic equation. Fromthen on, people de?ne that a pseudo-parabolic equation is an arbitrary higher-order partial di?erential equation with the ?rst-order derivative with respect to time. Inthis monograph, we study the case with the third order term ?ut.As a type of non-classical di?usion equations, pseudo-parabolic equations de-scribe a variety of important physical processes, such as the seepage of homogeneous?uids through a ?ssured rock [2] (where the coe?cient of third order term corre-sponds to a reduction in block dimension and an increase in the degree of ?ssuring),the unidirectional propagation of nonlinear, dispersive, long waves [3, 29] (where u istypically the amplitude or velocity), the aggregation of populations [22] (where u isrepresents the population density), the heat conduction involving a thermodynamictemperatureu ? k?u and a conductive temperatureu [59]. Specially, the followingpseudo-parabolic equation with sourcescan be used in the analysis of nonstationary processes in semiconductors in the pres-ence of sources[15, 16], where k ???tu ? ??ut corresponds to the free electron density rate,?ucorrespond to the linear dissipation of the free charge current and up describes asource of free electron current. From the physical point of view, the blow-up of solu-tions corresponds to electric breakdown in semiconductors or magnetic breakdownin magnetics, which occur in experiments.In the past several decades, pseudo-parabolic equations have been studied bymany scientists; some of them studied wave pseudo-parabolic equations, others—dissipative pseudo-parabolic equations. Nonlinear wave pseudo-parabolic equationsof third and ?fth orders were studied in [60, 61, 62, 63, 64, 65, 66, 67]. In thesepapers, initial value,initial-boundary value, and periodic problems were consideredand the global solvability and blow-up of solutions were analyzed. For globallysolvable equations, the asymptotic behavior of solutions was obtained, scatteringtheory was developed, and the stability of solitary-wave solutions was analyzed. For dissipative pseudo-parabolic equations, [30, 26, 9] investigated the initial-boundaryvalue problem and the Cauchy problem for linear pseudo-parabolic equation andestablished the existence and uniqueness of solutions. From then on, considerableattentions have been paid to the study of nonlinear pseudo-parabolic equations (seefor example [4, 5, 7, 11, 12, 17, 18, 23, 28] and the references therein). Not only theexistence and uniqueness results were obtained, but also the properties of solutions,such as asymptotic behavior and regularity, were investigated.With thorough research, people found that there are close connections andessential di?erences between pseudo-parabolic equations and parabolic equations.First, pseudo-parabolic equations are regularization of parabolic equations. Just asmentioned above, the solution of a parabolic equation can be obtained as a limitof solutions to the corresponding problem for pseudo-parabolic equations, this isthe reason that Showalter and Ting called it"pseudo-parabolic". In [26, 101], theypointed out that the solutions of pseudo-parabolic equations ??ut ?εM ??ut = Lu limitto the solution of parabolic equation ??ut = Lu in L2 sense, where M and L are self-adjoint elliptic operator. Though a number of regularizations have been proposed,include biharmonic regularizationε?2u, hyperbolic regularizationεutt etc, unlikethose regularized methods, pseudo-parabolic regularization has a unique advantage:well-posed initial-boundary value problem for parabolic equations are also well-posedfor pseudo-parabolic equation, namely, the de?nite conditions of parabolic equa-tions are enough to assure the well-posedness of pseudo-parabolic equations. Thisis another reason that Showalter and Ting called it"pseudo-parabolic". Second,the backward pseudo-parabolic equations are well-posed, then we can use back-ward problem of pseudo-parabolic equations to approximate backward problem ofparabolic equation. Take the backward heat equation for example, the problem satis?ed the ?nal condition u(x, T ) =χ(x) is unstable and not well-posed. But ifwe let the solution of backward equationsatis?ed v(x, T ) =χ(x)be the initial value of the above heat equation, thenεli→m0 uε(x, T ) =χ(x), see [104,105] for details. Recently, the forward-backward models used in image processing,biomathematics have attracted much attention, pseudo-parabolic regularization isa common method for analysis of forward-backward problem. Take the famous PMmodel in image processing for example,is essentially a forward-backward equation, [78] discussed the well-posedness ofpseudo-parabolic regularized equation. Besides, in general the hyperbolic equation?t2 =σ(ux)x (x∈R,σ(x) = 0), dose not posses global smooth solutions no matterhow smooth and small the initial data are. But the addition of the third order termuxxt will make the original problem more tractable. If the initial value is su?cientlysmall, then there exist a unique global smooth solution, see [58]. Although we canuse pseudo-parabolic regularization, there are essential di?erences between pseudo-parabolic equations and parabolic equations. From the results in [26], [9], [102] and[103], it can be seen that the regularity of the solution u for both u, ut and theinitial condition u(·, 0) belong to the same Sobolev space with respect to the spacevariable x. This feature is di?erent from problems with parabolic equations, wherethe regularity order of u is higher than that of ut with respect to x. Further, the ?nite-element methods analysis in [106] showed that the needed regularity of initialcondition is higher than the corresponding parabolic and lower than hyperbolic equa-tions. Thus, we can roughly speak that, the behavior of pseudo-parabolic equationsis intermediate between the behaviors of parabolic and hyperbolic equations.As mentioned above, the existence of third order term ?ut do not changethe de?nite conditions, and keep the regularity of initial data. Furthermore, ifwe consider other properties, such as long time behavior, is there any in?uence ofthird order term As everyone knows, the study on the problem with sourcesis an important part in the theoretical study on classical di?usion equations. Theappearance of sources a?ects the properties of solutions deeply, especially the longtime behavior of solutions, it may make solutions to blow up or extinguish in ?nitetime. In the results of Fujita [8] which considered Cauchy problem on heat equationwith interior sources, he found that the value of p has great e?ect on the solution,and it can describe the property of large time behavior of solutions accurately. Thesubsequent research according to this work gradually formed critical exponent theoryof classical di?usion equations. Then for the nonclassical di?usion equations withsources—–pseudo-parabolic equations, can we establish the corresponding criticalexponent theory? Will the critical exponent be in?uenced by the third order ?ut?As we know, we can ?nd the global existence and ?nite time blow-up results ofthe pseudo-parabolic equations with sources only in Levine [70], Kozhanov [68, 69],several papers of Korpusov et al ([71] and the references therein) and Kaikina etal [11, 72]. Kozhanov investigated the following initial-boundary value problem ofnonlinear Boussinesq equation with sourcesThrough constructing upper and lower solutions and using the comparison principle,he proved the following existence and nonexistence results (i)when p≥m≥2, then there exist global solutions if the initial values aresu?cient small;(ii)when p≥m≥2, then no nontrivial global solutions exist if the initial valuesare su?cient large.For the strong nonlinear pseudo-parabolic equations, especially for the case that themixed terms are nonlinear elliptic operators such asonly Korpusov et al have studied carefully, see the survey paper [70] and the refer-ences therein. It was Kaikina et al [11] who considered the superlinear case of theCauchy problemwith p > 1 and proved the existence and uniqueness of solutions using the integralrepresentation and the contraction-mapping principle. Furthermore, it was shownthat the Cauchy problem has a unique global solution ifσ> 2/n for u0 beingsu?ciently small. Subsequently, Kaikina [72] considered the initial-boundary valueproblem for (1) on a half-line and proved the global existence of solutions whenσ> 1 and the initial data is small enough. We conclude that in the existing results,except the work of Kaikina et al, most of them considered the case of general sourcesor special cubit sources, for the case of up, most considered initial-boundary valueproblem, and the work of Kaikina et al only restricted to the global existence.According to that, there is not any discussion about the critical exponent ofpseudo-parabolic equations, the existing results aimed at special cases, and thereis not a complete system. The main aim of this paper is to establish the criticalexponent theory of pseudo-parabolic equations with di?erent sources and in di?erentdomains to dill the theory gap. In this thesis, we discuss the following Cauchy problem of homogeneous(f(x)≡0)ï¿ inhomogeneous (f(x)≡0) semilinear pseudo-parabolic equationswhere p > 0, k > 0, u0(x)≥0, f(x)≥0 su?ciently smooth, and the Neumannboundary condition problemin exterior domains, where p > 1, k > 0, u0(x)≥0, f(x)≥0 su?ciently smooth,?≡Rn \ B1(0), Bl(0) the ball in Rn with radius l and centered at the origin, n theunit inner normal to the boundary of the unit ball, namely the unit external normalto the boundary of ?.Let us review some results on critical exponent of classical di?usion equationsrelated to our problem. It was Fujita who did the ?rst work about this problem,in [8] he considered Cauchy problem on heat equation with interior sources on Rn:ut = ?u + up, he proved the following:(i) if 1 < p < 1 + n2, then no nontrivial nonnegative global solutions exist;(ii) if p > 1 + n2, then there exist global positive solutions if the initial valuesare su?cient small.In the critical case p = 1 + 2/n, it was shown by Hayakawa [10] for dimensionsn = 1, 2 and Kobayashi et al [14] for all n≥1 that the problem possesses nonontrivial global solution u satisfying u(·, t) L∞(Rn) < +∞, t≥0. People call theabove result as Fujita type result, and pc = 1 + 2/n is said to be the Fujita critical exponent. Since Fujita critical exponent can describe the behavior of solutionsaccurately, such problems attract more and more people's attention. There havebeen a number of extensions of Fujita's results [8] in several directions, includingsimilar results from numerous of quasilinear parabolic equations and systems invarious of geometries (whole spaces, cones and exterior domains) with nonlinearsources and nonhomogeneous boundary value conditions. Bandle and Levine [73]considered the Fujita critical exponent of the equation ut = ?u + up with Dirichletboundary condition on D, which is a domain in Rn with boundary, and it was shownthat Fujita's statement held when D has bounded complement. When consideredthe Neumann zero boundary condition, Levine and Zhang [37] proved that the Fujitaexponent of the exterior space problem is still 1 + 2/n. It is remarkable that thecritical exponent is pc = 1 in the case that D is bounded domain. Recently, theauthors of [43, 45, 46, 47, 48] extended Fujita's result to inhomogeneous Cauchyproblems and found that the value of the critical exponent and the location of blow-up points are not the same as those for the homogeneous equation. The criticalexponent is more closely tied to the critical exponent of the corresponding ellipticalproblem. For example, the critical exponent n/(n ? 2) is the same as the in?mumof those p for which ?u + up = 0 in Rn has singular solutions of the form u =A|x|?2/(p?1). We refer to the survey papers [6, 19], the recent works [1, 20, 21, 25,32, 33, 34, 35, 36] and the references therein for a detailed account.In the ?rst chapter, we discuss the Cauchy problem and Neumann boundaryproblem of homogeneous pseudo-parabolic equations with nonlinear interior sources.For the Cauchy problem, just like the corresponding Cauchy problem of the semi-linear heat equation, there still exist tow critical exponents p0 = 1, pc = 1 + 2/n,such that(i)there exist global solutions for each initial datum in the case 0 < p≤p0,while there exists at least one initial datum such that the solution blows up in a ?nite time in the casep > p0;(ii)any nontrivial solution blows up in a ?nite time in the case p0 < p≤pc,while there exists at least one nontrivial global solution in the case p > pc.Here p0 and pc are called to be the critical global existence exponent and the criticalFujita exponent, respectively. For the Nuemann boundary problem, we prove thatthe Fujita exponent is still pc = 1+2/n. In the second chapter, we detailedly analyzethe Cauchy problem of semilinear pseudo-parabolic equations with inhomogeneousinterior sources and the Neumann boundary problem of semilinear pseudo-parabolicequations with inhomogeneous boundary condition. We prove the Fujita exponentsare pc = n/(n ? 2), this indicates that the appearance of inhomogeneous term makethe blow-up interval become larger.From the above results, we can easily discover that the appearance of third order?ut does not in?uence the magnitude of critical exponents. Seemingly there is littlesigni?cance of third order term, but if we refer to the explanation in Barenblatt et al[78]: third order term ?ut is a viscous term, has viscous relaxation e?ect, then ourresults review that the e?ect of this viscous term is not strong enough to change thecritical exponents, and will not in?uence the essence of the equation to change theproperties of the solutions, which shows the third order is a"good"viscous term.In addition, the estimation of the upper bound of the blow-up time shows that thethird order term will delay the blow-up time.The di?usion e?ect of the third order term does not inlude the results, butmakes it very di?culty to prove. First, it is almost impossible to use the methodconstructing supersolutions and subsolutions. We know that in studying critical ex-ponents for parabolic equations, the main methods is to construct global selfsimilarsupersolutions and blowing-up self-similar subsolutions, and most studies used thismethod. When there is no third order term, the construction of self-similar solutionscan be solved by the discussion on the corresponding ordinary di?erential equation or inequality. Since the order with irregularity brought by the third order term, theproblem we meet not only normal ordinary di?erential equation, but the ordinarydi?erential equation with parameter variable. Since sign rule is not suitable to thethird order term, then we could not scale as in [107] to make it a high order ODE.In addition, it is more di?cult to solve high order ODE than second order ODE.Then we need to ?nd some other useful methods for our research.Due to the main part of ??ut ? ???tu ? ?u is linear, then it is nature to use theexplicit form of Green function to ?nd the critical exponents, which is the usualmethod in critical exponent theory of heat equation with sources. Inspired by [11],we follow their method, use the integral representation and the contraction-mappingprinciple to prove the global existence results. When considering the exterior do-main problem, there is not any corresponding results about the Green function,fortunately, we can construct a global supersolution. In the study of inhomogeneousproblem, we use the stationary solution not related to the time to have the globalexistence.After the global existence, we turn to the ?nite time blow up. Though we canexpress the solution of the pseudo-parabolic equation explicitly by the fundamentalsolutionwhere the kernel functions G(x, t) and H(x, t) are more complicated than that of theheat equation and not able to write the concrete expression, so it is disable toprove blow up through estimating the fundamental solution. In this thesis, we usethe method to show the energy blowing-up, which was used in [24, 31, 36] to theCauchy problem of quasilinear parabolic equations including the semilinear parabolicequations. That is to say, we will show the energy blowing-up by determining theinteractions among the two kinds of di?usions and the sources via a series of preciseintegral estimates. However, due to the appearance of the third order term, the proofis more complicated, especially for the critical case p = pc, we need to initialize eachstep.In the above discussion, we need to establish the comparison principle of pseudo-parabolic equation, this is another di?culty in our research. The existing results areabout the special comparison principle for special pseudo-parabolic equation [69],which has great limitation, even can not have the uniqueness and nonnegative ofthe solution. Due to the appearance of the third order term, some conventionalmethods, such as sign rule, Holmgren method, are not useful. Here we give generalcomparison principle, for the Cauchy problem, we use the nonnegativity of the kernelfunction, and for the exterior domain problem, we use the maximum principle ofthe pseudo-parabolic equations [41, 11].We know that partial di?erential equations is originally derived from the prob-lems in the areas of physics. It is a branch of mathematics which has a very strongapplication background. People do research on the general theory of partial di?er-ential equations, while also continue to explore its potential applications. In recentyears, using partial di?erential equations for image restoration is its applied researchfocus.Let u be an original image, u0 the observed image, then we can use the de-graded model u0 = u +ηto describe noise interference, whereηstands for a white di?usion models [88]. On the basis of the above model, they gave some improvedmodels, among which the following model has caught out attentionwhere K is a threshold constant. From their discussion, (4) favors more preservationof edges. While one may ?nd when the noise is very large, (4) will be unstable inpresence of noise which is similar to that of the PM model.Inspired by the ideas of [98] and [94], we investigate several mixed type PDEsmodels in the last chapter. First, we investigate the following improved nonlinearparabolic-hyperbolic modelwhere Gσ(x) is the Gaussian kernel function, namelyAs we shall verify, our model is robust to the presence of the noise and improvesthe ability of identifying correctly the edge and it is also anisotropic di?usion. Itwill be seen that the numerical performances of the model are indeed better inthe very noisy cases. Indeed, it is very di?cult to con?rm the well-posedness inmathematics, which is a forward and backward equation essentially. We can proveexistence and uniqueness of the local solutions, which is very important for thenumerical computation.Second, we try to use hyperbolic-pseudo-parabolic equation models in image restoration ?rstly. We give the following three modelswhere n is the unit outer normal.Our experiments show that these three models achieve good results with denois-ing and boundary preserving at the same time. Hyperbolic model mechanism allowsthe enhance e?ect of our model, while the dissipation term ut and the third-orderterm ?ut make the model to avoid over-sharpening. In particular, the ?rst model ismore dominant in strengthening the image boundary than PM model, and overcomethe drawback of PM model in lefting large noise. The second model improve thepiecewise constant disadvantage of TV model. In preserving the boundary, the thirdmodel has very good results, while the third-order term ?ut make denoising e?ectvery evident. The numerical experiments of these models coincides with the factthat the strong damping term and the dissipation damping term would dissipate energy or prevent the accumulation of energy in the viscoelastic material movementprocess. The research of this part are still in the trial stage, the theoretical analysisof these models are our interest in the follow-up study.
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